Gracefulness of P - nLeggedC4- P' Graph

In this paper, we studied the Incident Binary Equivalent Decimal Edge Graceful Labeling (IBEDE) [1] and Strong Incident Binary Equivalent Decimal Edge Graceful Labeling (SIBEDE) [1] for Wheel Graph W1,n and Middle Graph of Wheel Graph M(W1,n). Also we obtained the Wheel Graph W1,n and its Middle Graph M(W1,n) for n ≥ 3 are IBEDE Graceful Labeling . And the Wheel Graph W1,n for n > 3and its Middle Graph M(W1,n) for n ≥ 3 are SIBEDE Graceful Labeling. Here the Ramsey number of IBEDE and SIBEDE Graceful Labeling for W1,5 is 3 and the Ramsey number of IBEDE and SIBEDE Graceful Labeling for M(W1,3) is 18.

Graceful graphs were first studied by Rosa in 1966. The Kotzig-Ringel graceful tree conjecture states that every tree has a graceful labeling. Aldred and McKay and others have used computer programs to show that trees of order up to 35 are graceful. Bagga et al. investigated algorithms for generating all graceful labelings of certain known classes of graceful graphs, including paths, cycles, and certain other classes of unicyclic graphs. The data generated by such algorithms has led to the discovery of new properties of such graceful labelings. In this paper we present a survey of graceful graph labeling algorithms and related complexity issues.DOI : http://dx.doi.org/10.22342/jims.0.0.14.1-9

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

Voluminous graph data management is a daunting problem in every real world application of a kind, besides the advancements in computation and storage technology. Efficient graph summarization techniques were contributed to achieve the substantial need for preserving novelty in social graphs. A GraceOutZip compression friendly graph reordering technique using graceful labeling strategy is adopted. User defined probabilistic selection method that provides unique labels for every identified outlier for potential use. Proposed method exploits unicycle-star based community representation rendering assignment of both node and edge labels based on graceful property. A novel mathematical programming model GraceOutZip is proposed to perform lossless compression with graph decomposition and unique label arrangement with intention for futuristic graph reconstruction. The experimental study on different real world network datasets demonstrates that GraceOutZip shows better scalable performance in perspective of interactive large-scale visual analytics and query optimization with 4 times better compression than the state-of-the-art representative method.

A graceful labeling of a graph G with ‘q’ edges and vertex set V is an injection f: VG → {0,1,2,….q} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |fu – fv| . A graph which admits a graceful labeling is called a graceful graph. A Shell graph is defined as a cycle Cnwith n -3 chords sharing a common end point called the apex . Shell graphs are denoted as Cn, n- 3. A multiple shell is defined to be a collection of edge disjoint shells that have their apex in common. Hence a double shell consists of two edge disjoint shells with a common apex.

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

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In graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

Professor Govind Swarup was a distinguished radio astronomer who drove the development of the field in India. Over a long career he built the Ooty Radio Telescope (ORT) and the Giant Metrewave Radio Telescope (GMRT), which are world-class facilities. Swarup was born in North India, had his undergraduate and post-graduate education at the University of Allahabad and then joined the National Physical Laboratory at New Delhi. He then travelled on a fellowship to Australia, to work on radio astronomy under J. L. Pawsey (FRS 1954). Later, in the USA, he completed his PhD with Ronald N. Bracewell at Stanford. Govind then accepted an offer made by Homi Bhabha FRS to begin radio astronomy work at the Tata Institute of Fundamental Research (TFIR). At TIFR, Govind conceived and built a novel, low cost, equatorially mounted cylindrical radio telescope, the ORT. The ORT was a versatile instrument and fuelled the growth of radio astronomy in India. In the late 1980s he conceived and led the construction of the GMRT, which again was based on a pioneering design. The GMRT is a remarkably successful instrument used by radio astronomers across the world. Govind Swarup has left behind a large group of radio astronomers who have continued his work and are finding new directions.

Given a partition of an n element set into equivalence classes, we study the problem of assigning unique labels to these elements in order to support the query that asks whether the elements corresponding to two given labels belong to the same equivalence class. This has various applications including for testing whether two vertices are in the same connected component in an undirected graph or in the same strongly connected component in a directed graph. We consider the problem in several models. Concerning labeling schemes where we assign labels to elements and the query is to be answered just by examining the labels of the queried elements (without any extra space): if each vertex is required to have a unique label, then we show that a label space of \(\sum _{i=1}^n \lfloor {n \over i} \rfloor \) is necessary and sufficient. In other words, \(\lg n + \lg \lg n + O(1)\) bits of space are necessary and sufficient for representing each of the labels. This slightly strengthens the known lower bound and is in contrast to the known necessary and sufficient bound of \(\lceil \lg n \rceil \) for the label length, if each vertex need not get a unique label. Concerning succinct data structures for the problem when the n elements are to be uniquely assigned labels from label set \(\{1,\ldots , n\}\), we first show that \(\varTheta (\sqrt{n})\) bits are necessary and sufficient to represent the equivalence class information. This space includes the space for implicitly encoding the vertex labels. We can support the query in such a structure in O(1) time in the standard word RAM model. We also develop a dynamic structure that uses \(O(\sqrt{n} \lg n)\) bits to support equivalence queries and unions in \(O(\lg n/\lg \lg n)\) worst case time or \(O(\alpha (n))\) expected amortized time where \(\alpha (n)\) is the inverse Ackermann function. Concerning succinct data structures for the problem when the n elements are to be uniquely assigned labels from label set \(\{1,\ldots , cn\}\) for any constant \(c > 1\), we show that \(\varTheta (\lg n)\) bits are necessary and sufficient to represent the equivalence class information. Moreover, we can support the query in such a structure in O(1) time in the standard word RAM model. We believe that our work can trigger further work on tradeoffs between label space and auxiliary data structure space for other labeling problems.

Given a partition of an n element set into equivalence classes, we consider time-space tradeoffs for representing it to support the query that asks whether two given elements are in the same equivalence class. This has various applications including for testing whether two vertices are in the same connected component in an undirected graph or in the same strongly connected component in a directed graph.We consider the problem in several models. Concerning labeling schemes where we assign labels to elements and the query is to be answered just by examining the labels of the queried elements (without any extra space): if each vertex is required to have a unique label, then we show that a label space of \(\sum_{i=1}^n \lfloor{n \over i} \rfloor\) is necessary and sufficient. In other words, \(\lg n + \lg \lg n + O(1)\) bits of space are necessary and sufficient for representing each of the labels. This slightly strengthens the known lower bound and is in contrast to the known necessary and sufficient bound of \(\lceil \lg n \rceil\) for the label length, if each vertex need not get a unique label. Concerning succinct data structures for the problem when the n elements are to be uniquely assigned labels from label set {1,…, n}, we first show that \(\Theta(\sqrt n)\) bits are necessary and sufficient to represent the equivalence class information. This space includes the space for implicitly encoding the vertex labels. We can support the query in such a structure in \(O(\lg n)\) time in the standard word RAM model. We then develop structures where the queries can be answered in \(O(\lg \lg n)\) time using \(O(\sqrt n \lg n/\lg \lg n)\) bits, and in O(1) time using \(O({\sqrt n} \lg n)\) bits of space. We also develop a dynamic structure that uses \(O(\sqrt n \lg n)\) bits to support equivalence queries and unions in \(O(\lg n/\lg \lg n)\) worst case time or O(α(n)) expected amortized time where α(n) is the inverse Ackermann function.KeywordsEquivalence ClassConnectivity QueryUnique LabelEquivalence QueryLabel SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Fertility decline in most countries of sub-Saharan Africa has thus far started later and proceeded more slowly than in countries in Asia and Latin America and the Caribbean undergoing the transition in fertility since the 1950s from high levels to near-replacement or even below-replacement levels (Bongaarts and Casterline 2013). Yet there is considerable variation among countries in sub-Saharan Africa: in the duration and magnitude of fertility decline, whether stalls in fertility decline have occurred, shifts in the timing of births, and even the economic and population subgroups that have led declines in family size (Caldwell, Orubuloye, and Caldwell 1992; Bongaarts and Casterline 2013; Cleland, Onuoha, and Timæus 1994; Cohen 1998; Ezeh, Mberu, and Emina 2009; Garenne 2008; Kirk and Pillet 1998; Rossier, Corker, and Schoumaker 2015; Timæus and Moultrie 2008). With current fertility estimated at 5.1 births per woman in the region and 19 countries in sub-Saharan Africa at or above that level and another 21 countries with at least four births per woman on average (United Nations 2015a), the pathways that future fertility takes will significantly determine population growth and age structure shifts not only in the region, but increasingly for the world. Sub-Saharan Africa is projected to grow from 840 million people in 2010 to nearly 1.4 billion in 2030 (United Nations 2015a). Above-replacement fertility is projected to account for 61 percent of this population increase from 2010 to 2030 compared to 4 percent from mortality reduction, 37 percent from a young age structure in 2010 (population momentum), and a small negative contribution from migration (United Nations 2015b). These projections draw on the United Nations medium variant and do not take into account the uncertainty around current and future fertility levels, uncertainty that only increases the farther the projection period extends. For high-fertility countries in sub-Saharan Africa, the wide uncertainty around where fertility is headed can result in substantial differences in population projections (Ezeh, Mberu, and Emina 2009; Fuchs and Goujon 2014; Gerland et al. 2014). Beyond population numbers alone, the uncertainty about fertility decline also bears on policy-relevant questions such as the degree to which the region may realize a demographic dividend (e.g., how fast the shift will occur toward a higher ratio of working-age population to non-working-age population and the consequent effects on economic growth) (Bloom et al. 2013) or the extent to which greenhouse gas emissions might be reduced by slowing population growth (O'Neill et al. 2010). Earlier reviews of the fertility transitions in sub-Saharan Africa through the 1980s and 1990s showed fertility declines underway in most countries and particularly rapid declines in several countries in Eastern Africa (Kenya, Rwanda, and Zimbabwe) and Southern Africa (Botswana and South Africa) (Cleland, Onuoha, and Timæus 1994; Cohen 1998; Kirk and Pillet 1998). Subsequent survey data suggested an apparent slowdown in the pace of fertility decline in more than ten sub-Saharan African countries (Bongaarts 2008). Further analyses of the data indicated far fewer stalls in fertility decline had in fact occurred in the region, with evidence strongly supportive of stalls in Kenya and Rwanda, and the stalls that do occur have been of relatively short duration (Garenne 2011; Machiyama 2010; Schoumaker 2009, 2014). Most countries in sub-Saharan Africa still lack complete and accurate vital registration data on births, so these and other analyses of fertility trends will continue to rely heavily on survey data and require reconciling estimates from different sources (Alkema et al. 2012; United Nations 2015c). Our aim in this chapter is to provide an updated and concise description of the diversity of fertility decline patterns among countries1 in sub-Saharan Africa, drawing on the latest series of fertility estimates that take into account many different data sources and that are harmonized with other demographic components (United Nations 2015d). We focus on the level of fertility prior to the start of fertility decline, the time period of the fertility transition, and the estimated pace of decline. We also explore the implications of different fertility decline patterns for future fertility and population projections in the region. We draw on the distinct patterns of fertility decline among countries worldwide that are advanced in (or have completed) their first fertility transition to construct probabilistic fertility and population projections for sub-Saharan African countries. The illustrative comparisons of projections highlight the demographic impact if future fertility decline in sub-Saharan countries were to accelerate and follow the rapid pace of decline already experienced by a diverse group of countries. The UN Population Division publishes estimates and projections of period total fertility rates in World Population Prospects (WPP) every two years. The estimates of total fertility presented in this chapter are from the 2015 Revision and are for five-year time periods for countries or areas with 90,000 persons or more in 2015 (United Nations 2015a). The most recent data underlying the total fertility estimates from the 2015 Revision for 50 sub-Saharan African countries2 (United Nations 2015c) are from the period 2013–2014 for 14 countries, 2010–2012 for 30 countries, and 2005–2009 for six countries. A common challenge in estimating total fertility over time, especially for countries without accurate or complete vital registration data,3 is that estimates vary across data sources and by the methodology used to derive those estimates. Schoumaker (2014) showed that the underlying data from standardized, high-quality surveys such as the Demographic and Health Surveys vary considerably within and across countries, yielding total fertility estimates from recent fertility data of good quality (e.g., Gabon, Lesotho, Namibia, and Zimbabwe) and of poor quality (e.g., Benin, Burkina Faso, Cameroon, Chad, Ethiopia, Guinea, Madagascar, Mali, Mozambique, Niger, Nigeria, and Uganda). Total fertility estimates based on births in the last three years tend to be under-estimated by 10 percent or more in most of the surveys with poor quality fertility data from retrospective birth histories. Figure 1 illustrates the variation in total fertility estimates based on survey data and estimation methods (direct methods (D) and cohort-completed (C) fertility) for Nigeria for the period 1985 to 2015. The thick lines show the total fertility estimates from the 2010, 2012, and 2015 Revisions of WPP. Given new data from the 2008 DHS and other surveys, total fertility in the 2012 Revision was re-estimated at a higher level than the 2010 Revision beginning in the mid-1980s, resulting in almost half a birth per woman difference in the 2005–2010 period. The 2015 Revision used new data from the 2013 Demographic and Health Survey. The 2013 survey estimates highlight a recurring pattern in which fertility estimates based on a recent reference period are consistently lower than fertility estimates from reconstructed birth histories for the same time point. Looking only at fertility estimates from a three-year reference period, the 2013 DHS shows a decline in total fertility to 5.5 births per woman from a stalling pattern of 5.7 births per woman in the 2003 and 2008 DHS. Yet the absolute differences are large between these three-year reference period estimates and those for the same time point from the reconstructed birth histories: about half a birth difference in the mid-2000s (comparing the 2008 and 2013 survey estimates) and about one birth difference in the early 2000s (comparing the 2003 survey estimate to those from the 2008 and 2013 surveys). Estimates of the total fertility rate for Nigeria 1985–2015 based on various data sources and estimation methods, and WPP estimates from the 2010, 2012, and 2015 Revisions NOTES: DHS = Demographic and Health Survey; MICS = Multiple Indicator Cluster Survey; MIS = Malaria Indicator Survey. (C) refers to cohort completed fertility (i.e., average number of children ever born) for women aged 40–44 and 45–49 at the date of the survey and backdated using their mean age of childbearing. (D) refers to direct fertility estimates based on maternity histories or recent births in the 12 or 24 months preceding the survey. SOURCES: Federal Office of Statistics of Nigeria (1992); National Bureau of Statistics (2008, 2012); National Population Commission (2000, 2002, 2004, 2009, 2012, 2014). WPP fertility estimates consider as many types and sources of empirical estimates as possible, including retrospective birth histories and direct and indirect fertility estimates (Gerland 2014). The 2015 Revision updated all total fertility estimates taking into account new data and the inconsistencies among estimates. Moreover, total fertility estimates are derived to ensure as much internal consistency as possible with all other demographic components and intercensal cohorts enumerated in successive censuses (United Nations 2015d). The advantages of this approach are that the estimates are internally consistent within a country over time and with respect to other related demographic information. A disadvantage is that the estimates can depart from a country's official estimates of fertility. Figure 2 shows the estimated trends in period total fertility for sub-regions of sub-Saharan Africa from 1950 to 2015. It was high (above six births per woman) in all sub-regions in 1950–1955. Fertility remained high in Eastern and Western Africa until the 1980s, after which it began a slow decline to 4.9 births per woman in Eastern Africa and 5.5 in Western Africa in 2010–2015. Fertility in Middle Africa began to decline a decade later and more slowly, reaching 5.8 births per woman in 2010–2015. Southern Africa departed from the overall trends with a decline beginning in the 1950s and dropping below three births per woman in the 2000s. The 2010–2015 estimate of 2.5 births per woman in Southern Africa is about half the total fertility level in Eastern, Middle, and Western Africa. Sub-regional trends in total fertility, sub-Saharan Africa, 1950–2015 SOURCE: United Nations 2015a The sub-regional fertility levels mask diverse levels among countries. Figure 3 shows country-specific total fertility levels in 2010–2015. Among the 16 countries in Western Africa, total fertility ranged from 2.4 in Cabo Verde to 7.6 in Niger. One additional country had a fertility level of more than six births per woman (Mali), six countries had fertility between five and six births per woman (Burkina Faso, Côte d'Ivoire, Gambia, Guinea, Nigeria, and Senegal), and seven had fertility between four and five births per woman. Total fertility levels in countries in Africa, 2010–2015 SOURCE: United Nations 2015a. Total fertility levels ranged even more widely among the 20 countries in Eastern Africa, from 1.5 in Mauritius to 6.6 in Somalia. One additional country in Eastern Africa still had a fertility level above six in 2010–2015 (Burundi) and six countries had fertility between five and six births per woman (Malawi, Mozambique, South Sudan, Uganda, Tanzania, and Zambia). Eight countries had fertility levels between four and five births per woman. The lowest levels of fertility were in Djibouti (3.3 births per woman) and in the small island countries of Mauritius, Réunion, and Seychelles (less than three births per woman). The nine countries of Middle Africa all had fertility levels at or above four children per woman. In Angola, Chad, and DR Congo, fertility levels were six or more births per woman, and in the remaining six countries fertility levels were between four and five births per woman. While fertility in Southern Africa is largely dominated by South Africa's pattern, the range in fertility among the five countries in the sub-region is narrow, from 2.4 births per woman in South Africa to 3.6 in Namibia. Both Botswana and South Africa now have fertility levels below three births per woman. The start of the fertility transition also varies widely across sub-Saharan Africa. We analyze when and at what level of fertility a country experienced a maximum total fertility level before the onset of fertility decline. This maximum is defined as the most recent five-year time period where total fertility is within half a child of the overall maximum fertility in the country over the 1950–2015 estimation period, thus excluding random fluctuations in pre-transition fertility (Alkema et al. 2011). Figure 4 shows the diversity across countries and within sub-regions in the level and time period of the maximum fertility level before the onset of fertility decline, as assessed in the 2015 Revision. By the late 1970s, 29 sub-Saharan countries were on the verge of a fertility decline, increasing to 40 countries by the early 1980s. While the maximum fertility before the onset of fertility decline was reached in all countries in Southern Africa by the late 1970s, the range of experiences was much wider among countries in Eastern Africa (from the early 1950s in Réunion to the late 1990s in Somalia), Middle Africa (from the late 1960s in Angola to the late 1990s in Chad and DR Congo), and Western Africa (from the early 1960s in Cabo Verde to the late 1990s in Niger). Maximum total fertility in the time period before the onset of fertility transition, sub-Saharan African countries by sub-region SOURCE: United Nations 2015a. The maximum fertility before the onset of fertility transition ranged from less than six births per woman in five countries (Central African Republic, Equatorial Guinea, Gabon, Lesotho, and Seychelles) to more than eight births per woman in two countries (Kenya and Rwanda). There was a positive but weak relationship between the maximum fertility level and the timing of when fertility transition commenced (R2 = .04). The transition from the maximum fertility to the current estimated fertility level in 2010–2015 has been slow for most countries in sub-Saharan Africa. We examine both the maximum decline in fertility in a five-year period and the duration of time between the maximum fertility level and when a country achieved a 10 percent decline in total fertility. A 10 percent decline in total fertility is one of several empirical rules that researchers have used to identify when fertility has begun to decline in a sustained manner from a pre-transitional maximum (Bongaarts and Casterline 2013; Coale and Treadway 1986; United Nations 2014), including differences in the time period selected to identify the onset of fertility decline (Casterline 2001) and the magnitude of decline (e.g., 5 percent decline with further conditions for subsequent changes applied, see Bryant 2007). While the different rules all strive to distinguish sustained decline from random fluctuations in fertility levels, each rule will have different repercussions for interpretations about the timing of onset of fertility decline, duration and pace of decline, and correlations with levels of development and other indicators. In seven countries the first steps of fertility decline took place over a long time period. In Angola, Gambia, and Uganda, a 10 percent decline from the maximum fertility level took 40 years; and in Lesotho, Mozambique, Niger, and Tanzania, it took 30 to 35 years (Appendix Table 1).4 In other countries the fertility decline never gained speed. Overall declines from the country-specific maximum fertility level to the level in 2010–2015 were very slow in nine countries (Angola, Congo, Gambia, Mali, Mozambique, Niger, Nigeria, Uganda, and Tanzania), where average fertility declines were 0.2 children per woman or less, a pace at which it would take at least 25 years to realize a decline of one birth per woman (Appendix Table 1). During the fertility transition, some countries experience an acceleration of fertility decline that, based on historical experience of past transitions in Latin America and the Caribbean and Asia, might reach a decline of more than one birth per woman in a five-year period (United Nations 2015a). In most countries in sub-Saharan Africa, however, the maximum fertility decline is much smaller. The largest declines of more than one birth per five-year period were registered in the early-transition countries in Eastern Africa before 2000 (Djibouti, Mauritius, Mayotte, Réunion, Rwanda, Seychelles, and Zimbabwe). For some countries the maximum fertility decline is projected to be reached in the future: one country in Eastern Africa (Mozambique), four in Middle Africa (Angola, DR Congo, Equatorial Guinea, and Sao Tome and Principe), and five in Western Africa (Gambia, Guinea, Mali, Niger, and Nigeria). Figure 5 shows that countries reaching the maximum fertility decline later tend to reach it at lower levels (the correlation across the 50 countries is R2 = .45). Maximum five-year decline in total fertility since the onset of fertility transition, actual and projected, sub-Saharan African countries by sub-region SOURCE: United Nations 2015a. The pace of fertility decline is illustrated in Figure 6 for Ethiopia and Nigeria, the two most populous countries in sub-Saharan Africa. Estimated levels of total fertility from 1950–1955 to 2010–2015 are shown with a fitted model of the fertility transition. Both countries reached maximum fertility levels before the onset of fertility decline between the late 1970s and early 1980s, and both reached a 10 percent decline from this maximum in the early 2000s. While Ethiopia reached the peak pace of fertility decline in 2005–2010 (a decline of nearly one birth in the five-year period), Nigeria's fertility decline has been consistently slow, with a low peak pace of decline. The maximum fertility decline per five-year period in Nigeria is projected to be reached in the future (a decline of 0.3 births in 2020–2025). The difference in the pace results in current total fertility of 4.6 births per woman in Ethiopia and 5.7 in Nigeria. The 80 percent prediction intervals (dashed lines in Figure 6) around the probabilistic projections of total fertility indicate the magnitude of the different fertility changes that could reasonably happen. For example, by 2045–2050 there is a one in ten chance that total fertility in Nigeria could be as low as 2.6 or as high as 4.4 (Figure 6 and Appendix Table 2). Observed decline in total fertility from 1950–1955 to 2010–2015, predicted decline, and time period of maximum TFR and maximum decline, Ethiopia and Nigeria SOURCES: United Nations 2015a and computations by authors. Given the slow pace of fertility decline in most sub-Saharan African countries thus far and the uncertainty around the pace of future declines, what might be the impact on future fertility levels and the growth in total population if fertility declines in the region followed an accelerated pace that has already been experienced by countries that have completed or are nearing completion of the fertility transition? To answer this question, we generated probabilistic fertility scenarios for 2015–2100 using a modeling approach described in Alkema et al. (2011) and implemented in a publicly accessible software package BayesTFR (Ševčíková, Alkema, and Raftery 2011). The standard United Nations 2015 probabilistic results pool the experience of all countries having similar fertility levels and trends (the baseline scenario). An alternative probabilistic scenario was created using only the fertility transition experiences of 21 countries5 that shared a similar pattern of accelerated decline in fertility (the accelerated scenario): specifically, a slow pace at the start of the transition that sharply rises to a peak pace of decline before steadily tapering off after total fertility has reached about four births per woman. These 21 countries, which include—among countries with more than 10 million inhabitants in 2015—Bangladesh, China, El Salvador, Morocco, Peru, South Africa, Sri Lanka, Thailand, Turkey and Uzbekistan, represent disparate institutional, social, economic, and cultural contexts and yet experienced a similar pattern of a relatively abrupt acceleration of fertility decline. The objective of this exercise is to examine the impact on population size of fertility decline that is more rapid than currently projected, and to illustrate the implications of such a hypothetical scenario. While we do not theorize or analyze the factors that produced these rapid fertility declines, the fact that a similar pattern of fertility decline took place within such a diverse group of countries raises the possibility that a similar decline could occur within a region that also reflects quite distinctive and diverse contexts. Each fertility scenario is used to simulate 10,000 probabilistic population projections for 2015–2100 under identical conditions (i.e., using the same mortality and migration assumptions) that show the population growth trajectories if sub-Saharan African countries were to follow a specific fertility decline pattern.6 The implications of this specific fertility decline pattern for future fertility trends are shown in Figure 7 for Ethiopia and Nigeria. The accelerated fertility decline scenario leads to much more rapid declines for these countries. By 2045–2050, median projected total fertility in Ethiopia would decline from 2.3 (baseline) to 2.0 (accelerated decline) and in Nigeria from 3.6 (baseline) to 2.6 (accelerated decline) (see Appendix Table 2). Probabilistic fertility projections (median and 80 percent prediction intervals) for two scenarios: baseline and accelerated fertility decline, Ethiopia and Nigeria SOURCES: United Nations 2015a and computations by authors. The accelerated fertility scenario has greater implications for projected fertility in Nigeria, where the recent pace of fertility decline has been much slower than in Ethiopia. The projected medians of the accelerated scenario are similar to the lower bound of the baseline scenario for parts of the projections, suggesting that Ethiopia and Nigeria would need to experience a rapid fertility decline, similar to the rapid declines experienced by the group of 21 countries, in order to realize the lower-bound fertility level of the baseline scenario. The assumption that Ethiopia and Nigeria would follow the fast fertility decline experiences of these 21 countries leads to substantially lower population projections compared with the baseline scenario (Figure 8 and Appendix Table 3). If Ethiopia experienced an accelerated fertility decline, the projected population grows from 99 million in 2015 to 182 million by 2100 (80 percent prediction interval of 93 million to 324 million) compared with the higher median projection of 234 million under the baseline scenario. If Nigeria adopted the accelerated fertility decline pattern, the projected population grows from 182 million in 2015 to a median of 466 million by 2100 (80 percent prediction interval of 282 million to 777 million) rather than the faster projected growth to 737 million under the baseline scenario. Probabilistic population projections (median and 80 percent prediction intervals) for two scenarios: baseline and accelerated fertility decline, Ethiopia, Nigeria, and sub-Saharan Africa SOURCES: United Nations 2015a and computations by authors. The counterfactual of sub-Saharan African countries following an accelerated fertility decline, as experienced by the group of 21 countries, results in a projected increase in total population in the region from 962 million in 2015 to 3.2 billion by the end of the century (with 80 percent probability of being between 2.8 and 3.7 billion). The total population projection for the baseline UN 2015 scenario would lead to a median projection of about 4 billion, or a difference of 770 million people with an 80 percent prediction interval mostly above the upper bound of the accelerated fertility decline scenario. We presented here an updated description of fertility decline in sub-Saharan Africa and have explored the implications for fertility and population projections if sub-Saharan African countries follow a pattern of accelerated fertility decline. Because estimates of fertility in the region rely almost entirely on survey and census data, there is still uncertainty about fertility change over time. Our analyses of trends over time were based on a new historical time series of total fertility estimates for the period 1950 to 2015 from the 2015 Revision of World Population Prospects. Fertility has remained persistently high in Eastern, Middle, and Western Africa (4.9 births per woman or higher as of 2010–2015) and declined rapidly in Southern Africa, which currently has about half the total fertility level of other sub-regions births per woman). Southern Africa is dominated by South Africa, TFR of 2.4 is the lowest in the Yet there is wide variation in total fertility across countries, even within from below fertility births per woman) in Mauritius to 7.6 births per woman in Niger. The fertility level before the onset of decline was high in all countries in sub-Saharan Africa, from less than six births per woman in five countries to more than eight births per woman in two countries. By the late 1970s, 29 countries were on the verge of fertility decline, increasing to 40 countries by the early 1980s. The transition from these maximum levels of fertility prior to the onset of fertility decline to current levels has been slow for most countries, including seven in which it took 30 years or more to a 10 percent decline in total fertility. While countries in Asia and in Latin America and the Caribbean experienced an acceleration in fertility decline of more than one birth per five-year time period, this level of acceleration has been experienced thus far by only seven countries in Eastern Africa before the The pace of fertility decline in sub-Saharan Africa will a large in the magnitude of future growth in We showed through an illustrative scenario that an acceleration in the pace of fertility decline to that already experienced by 21 countries South Africa) would the growth in future population in the region from a projected billion by to billion and total population size from a projected 4 billion people by the end of the century to 3.2 While these 21 countries represent a wide range of institutional, social, economic, and cultural a similar fast decline from fertility levels of six or more children per woman in the past to less than two or three in recent years. this accelerated pace of fertility decline to countries at of transition illustrates the substantial impact on future population growth of other possible patterns of fertility decline in sub-Saharan Africa. of this are assumption that the fertility estimates from the World Population Prospects are and the fact that uncertainty around the estimates of period total fertility in recent is not into By scenarios on the distinct fertility decline patterns that have been experienced thus far by countries that have completed or are advanced in their fertility we new patterns of fertility decline in the Because most countries in Middle Africa and Western Africa are still in the early of fertility transition, an sub-Saharan African pattern of fertility decline may still and Timæus The is not for the or of by the authors. than be to the for the

Cartography was a model for Dennis Oppenheim’s territorial conception of sculpture and the road map was a site on which he sought to expand sculpture’s boundaries. This article focuses on five works conceived in the late 1960s in which Oppenheim built upon the proprietary claim to space implicit in cartography. Whether plotted on a map or constructed alongside the highway, Oppenheim viewed these sculptures as instruments of spatial orientation and territorial possession, as well as mechanisms to reroute the infrastructures and informational networks of everyday life. In Oppenheim’s sculpture, the liberatory aesthetics of minimalism’s phenomenology is marked with a territorial violence that plays out in cartographic and real space. He sought a way to force the body to register within abstract systems and, in turn, to imprint those abstract systems on the body. The road map was both a model and a site for this exchange.

Research in graph theory has lead to one of the important area called labeling of graphs. There are different types of labeling such as graceful labeling, magic labeling, edge-graceful labeling, prime labeling, radio labeling, harmonious labeling etc., In this paper, we investigate the existence of edge graceful labeling for the Corona graph Cn⊙K2¯ (n is odd n ≥ 3). Also we have defined a flower graph which does not admit an edge-graceful labeling.