SYMMETRIC GRAPHS WITH 2-ARC TRANSITIVE QUOTIENTS

Constructing a Class of Symmetric Graphs

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -- which asks for a subgraph with minimum degree that connects a given set of vertices -- is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.The standard techniques for solving degree-bounded problems often fall in the category of iterative and dependent rounding techniques. Unfortunately, these rounding methods are inherently difficult to adapt to an online settings since the underlying fractional solution may change dramatically in between the rounding steps. Indeed, this might be the very reason that despite many advances in the online network design paradigm in the past two decades, the natural family of degree-bounded problems has remained widely open.In this paper, we design an intuitive greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Furer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach and its dual analysis, paves the way for solving the online variants of the classical problems in this family of problems.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Let G be a simple connected graph with a set of vertices V(G) and set of edges E(G). The distance from vertex u to vertex v on graph G is the length of the shortest path from vertex u to vertex v, which is denoted by d(u, v). Suppose there is an ordered set W = {w1, w2, w3,…,wn} and v is a vertex on graph G, then the representation of vertex v to W is an n - tuple ordered pair, namely r(v|W) = (d(v, w1),d(v, w2),d(v,w3),…, d(v, wn)). A set W is called a local resolving set if r(u|W) ≠ r(v|W) for each pair of vertex u and vertex v adjacent to each other on graph G. The number of vertices in the local metric base of graph G is called the local metric dimension and is denoted by lmd(G). Dipyramidal graph DPn is the skeleton of an n-sided dipyramid with n ≥ 3. King graph Kmn is a graph with m × n vertices in which each vertex represents a square in an m × n chessboard, and each edge corresponds to a legal move by a king with m, n ≥ 2. In this paper, the local metric dimension of a dipyramidal graph and a king graph were explained.

Let $G$ be a finite group and ${\rm\Gamma}$ a $G$-symmetric graph. Suppose that $G$ is imprimitive on $V({\rm\Gamma})$ with $B$ a block of imprimitivity and ${\mathcal{B}}:=\{B^{g};g\in G\}$ a system of imprimitivity of $G$ on $V({\rm\Gamma})$. Define ${\rm\Gamma}_{{\mathcal{B}}}$ to be the graph with vertex set ${\mathcal{B}}$ such that two blocks $B,C\in {\mathcal{B}}$ are adjacent if and only if there exists at least one edge of ${\rm\Gamma}$ joining a vertex in $B$ and a vertex in $C$. Xu and Zhou [‘Symmetric graphs with 2-arc-transitive quotients’, J. Aust. Math. Soc. 96 (2014), 275–288] obtained necessary conditions under which the graph ${\rm\Gamma}_{{\mathcal{B}}}$ is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to ${\rm\Gamma}$ and ${\mathcal{B}}$ and show that there is a unique graph corresponding to this case.

Ordered Ramsey numbers

Our aim is to prove some results concerning the connectivity, lineconnectivity and J-connection or connection modulo J of total graphs. Only finite, nonoriented graphs without loops or multiple edges will be considered. Let G be a graph whose vertex set is V(G) and whose edge set is E(G). The elements of the set V(G)wE(G) will be called the elements of G and two elements of G are said to be associated if they are either adjacent or incident. The total graph T(G) of G is a graph whose vertex set is V(G)uE(G), two vertices being joined by an edge if and only if they are associated elements of G (see [1]). As an example of a graph G and its total graph T(G) see Fig. 1. We make an obvious distinction: small rings represent point-vertices of T(G) (x is a point-vertex x ~ E(G)). T(G) contains both G and its line-graph or interchange graph [4] L(G) as disjoint subgraphs. Remember that by definition, V[L(G)] =E(G) and two vertices of L(G) are linked by an edge if and only if the corresponding edges of G are adjacent. Edges of T(G) belonging neither to G nor L(G) form what will be called the incidence-graph I(G) of G. By definition the connectivity k(G) of G is the least number of vertices whose removal disconnects G or reduces G to a single vertex; a set of k(G) vertices satisfying this condition is called a minimal separating vertex set of G. Moreover G is n-connected if and only if k(G) >= n. On the

We want to find the vertex sets of components of a graph G with a known vertex set V and unknown edge set E. We learn about G by sending an oracle a query set $S \subseteq V$\hspace*{-1pt}, and the oracle tells us the vertices connected to S. The objective is to use the minimum number of queries to partition the vertex set into components. The problem is also known as interconnect diagnosis of wiring networks in VLSI. We present a deterministic algorithm using O(min{k, lg n}) queries and a randomized algorithm using expected O(min{k,lg k + lg lg n}) queries, where n is the number of vertices and k is the number of components. We also prove matching lower bounds.

We study graphs with the vertex set ?2$\mathbb Z^{2}$ which are subgraphs of the 8-adjacency graph and have the property that certain natural cycles in these graphs are Jordan curves, i.e., separate ?2$\mathbb Z^{2}$ into exactly two connected components. Of these graphs, we determine the minimal ones and study their quotient graphs. The results obtained are used to prove digital analogues of the Jordan curve theorem for several graphs on ?2$\mathbb Z^{2}$. Thus, these graphs are shown to provide background structures on the digital plane ?2$\mathbb Z^{2}$ convenient for studying digital images.

Longest paths and cycles in faulty star graphs

Given an undirected graph G and two vertex subsets H 1 and H 2, the smallest bi-level augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertex-disjoint paths between every pair of vertices in H 1 and two edgedisjoint paths between every pair of vertices in H 2. We solve the bi-level augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log2 n) time using n + m processors on an EREW PRAM.KeywordsCritical NodeGraph ProblemVertex SubsetConnectivity RequirementAugmentation ProblemThese 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.

A graph is denoted by $G$ with the vertex set $V(G)$ and the edge set $E(G)$.A path $P=\langle v_0,v_1,\cdots,v_m\rangle$ is a sequence of adjacent vertices. Two paths with equal length $P_1=\langle u_1,u_2,\ldots,u_m\rangle$ and $P_2=\langle v_1,v_2,\ldots,v_m\rangle$ from $a$ to $b$ are {\it independent} if $u_1=v_1=a$, $u_m=v_m=b$, and $u_i\neq v_i$ for $2\leq i\leq m-1$.Paths with equal length $\{P_i\}^{n}_{i=1}$ from $a$ to $b$ are {\it mutually independent} if they are pairwisely independent. Let $u$ and $v$ be two distinct vertices of a bipartite graph $G$, and let $l$ be a positive integer length, $d_G(u,v)\le l \le |V(G)-1|$ with $(l-d_G(u,v))$ being even. We say that the pair of vertices $u$, $v$ is {\it $(m,l)$-mutually independent bipanconnected} if there exist $m$ mutually independent paths $\{P_i^l\}_{i=1}^m$ with length $l$ from $u$ to $v$. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices $u$ and $v$ in the $n$-dimensional hypercube, with $d_{Q_n}(u,v) \ge n-1$, is $(n-1,l)$-mutually independent bipanconnected for every $l$, $d_{Q_n}(u,v) \le l \le |V(Q_n)-1|$ with $(l-d_{Q_n}(u,v))$ being even. As for $d_{Q_n}(u,v) \le n-2$, it is also $(n-1,l)$-mutually independent bipanconnected if $l \ge d_{Q_n}(u,v)+2$, and is only $(l,l)$-mutually independent bipanconnected if $l=d_{Q_n}(u,v)$.

Let G = (V,E) be a simple connected molecular graph. The eccentric connectivity index ξ(G) is a distance–based molecular structure descriptor that was recently used for mathematical modelling of biological activities of diverse nature. In such a simple molecular graph, vertices represent atoms and edges represent chemical bonds, we denoted the sets of vertices and edges by V = V(G) and E = E(G), respectively. If d(u,v) be the notation of distance between vertices u,v ∈ V and is defined as the length of a shortest path connecting them. Then, the eccentricity connectivity index of a molecular graph Gis defined as ξ(G) = Σv∈V(G) deg(V)ec(V), where deg(V) (or simply dv) is degree of a vertex V ∈ V(G), and is defined as the number of adjacent vertices with V. ec(V) is defined as the length of a maximal path connecting to another vertex of v. In this paper, we establish the general formulas for the eccentricity connectivity index of molecular graphs classes of chemical trees with application to alkynes.

A (2, 1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0, 1, ..., k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2, 1)-total labeling number λ2T(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ2T(G) ≤ Δ(G)+2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2T(G) ≤ Δ(G)+2 even in the case of Δ(G) ≤ 4.

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.