Geodetic Sets in Graphs

On the geodetic number and related metric sets in Cartesian product graphs

Fiber-complemented graphs — I: structure and invariant subgraphs

A [Formula: see text]-design over regular graph [Formula: see text] of degree [Formula: see text] is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of geodetic sets in [Formula: see text] called blocks such that two vertices [Formula: see text] and [Formula: see text] which are [Formula: see text] associates occur together in [Formula: see text] blocks, the numbers [Formula: see text] being independent of the choice of the pair [Formula: see text] and [Formula: see text]. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from geodetic sets in some classes of graphs. Also, we give a complete list of PBIB-designs with respect to the geodetic sets for cubic graphs of order at most [Formula: see text]. The discussion of non-existence of some designs corresponding to geodetic sets from certain graphs concludes the paper.

Given a set of graphs, the median graph has been theoretically presented as a useful concept to infer a representative of the set. However, the computation of the median graph is a highly complex task and its practical application has been very limited up to now. In this work we present a new genetic algorithm for the median graph computation. A set of experiments on real data, where none of the existing algorithms for the median graph computation could be applied up to now due to their computational complexity, show that we obtain good approximations of the median graph. Finally, we use the median graph in a real nearest neighbour classification showing that it leaves the box of the only-theoretical concepts and demonstrating, from a practical point of view, that can be a useful tool to represent a set of graphs.

An independent set S of vertices in a graph G is an independent position set if no three vertices of S lie on a common geodesic. An independent position set of maximum size is an ip-set of G. The cardinality of an ip-set is the independent position number, denoted by ip(G). In this paper, we introduce and study the independent position number of a graph. Certain general properties of these concepts are discussed. Graphs of order n having the independent position number 1 or n − 1 are characterized. Bounds for the independent position number of Cartesian and Lexicographic product graphs are determined and the exact value for Corona product graphs are obtained. Finally, some realization results are proved to show that there is no general relationship between independent position sets and other related graph invariants

Many graph-theoretic concepts have both vertex and edge versions. Examples are cut-vertex , cut-edge , vertex-integrity ( integrity ), edge-integrity , vertex coloring , edge coloring , and vertex-connectivity , edge-connectivity . Frank Harary et al. defined the geodetic number of a graph for vertices in [Chartrand, G., Harary, F. and Zhang, P. (2002). On the geodetic number of a graph. Networks , 39 , 1-6; Chartrand, G., Harary, F. and Zhang, P. (2000). Geodetic sets in graphs. Discussions Mathematicae Graph Theory , 20 , 129-138; Harary, F., Loukakis, E. and Tsours, C. (1993). The geodetic number of a graph. Mathl. Comput. Modelling , 17 (11), 89-93]. In this study we give a definition of the edge geodetic number for a graph and derive some results.

A vertex set D in a graph G is called a geodetic set if each vertex of G is lying on some shortest u−v path of G, where u, v∈D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this article, we find out the geodetic numbers, the lower and the upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori.

The complexity of Dominating Set in geometric intersection graphs

An axiomatic approach to location functions on finite metric spaces

In this paper we consider two existing methods to generate a representative of a given set of graphs, that satisfy the following two conditions. On the one hand, that they are applicable to graphs with any kind of labels in nodes and edges and on the other hand, that they can handle relatively large amount of data. Namely, the approximated algorithms to compute the Median Graph via graph embedding and a new method to compute the Barycenter Graph. Our contribution is to give a new algorithm for the barycenter computation and to compare it to the median Graph. To compare these two representatives, we take into account algorithmic considerations and experimental results on the quality of the representative and its robustness, on several datasets.KeywordsRandom GraphGraph DatabaseMedian GraphGraph DomainGraph EmbeddingThese 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.

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number $\gamma_P(G)$ of $G$ is the minimum cardinality of a power dominating set. Dorfling and Henning [Discrete Appl. Math., 154 (2006), pp. 1023-1027] determined the power domination number of the Cartesian product of paths. In this paper the power domination number is determined for all direct products of paths except for the odd component of the direct product of two odd paths. For instance, if $n$ is even and $C$ a connected component of $P_m\times P_n$, where $m$ is odd or $m\geq n$, then $\gamma_P(C)=\left\lceil n/4 \right\rceil$. For the strong product we prove that $\gamma_P(P_n \boxtimes P_m) = \max\{\lceil n/3\rceil, \lceil (n+m-2)/4\rceil\}$, unless $3m-n-6 \equiv 4\pmod 8$. The power domination number is also determined for an arbitrary lexicographic product.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

A graph G is k-factor-critical if G ? S has a perfect matching for any k-subset S of V(G). In this paper, we investigate the factor-criticality in Cartesian products of graphs and show that Cartesian product of an m-factor-critical graph and an n-factor-critical graph is $${(m+n+\varepsilon )}$$ -factor-critical, where $${\varepsilon = 0}$$ if both of m and n are even; $${\varepsilon = 1}$$ , otherwise. Moreover, this result is best possible.

Maximum weight independent sets and cliques in intersection graphs of filaments

A dominating set is a classic concept that is widely used in road safety, disaster rescue operations, and chemical graphs. In this paper, we introduce a variation of the dominating set: the proper 3-dominating set. For a proper 3-dominating set D of graph G, any vertex outside D is adjacent to at least three vertices inside D, and there exists one vertex outside D that is adjacent to three vertices inside D. For graph G, the proper 3-domination number is the minimum cardinality among all proper 3-dominating sets of G. We find that a graph with minimum degree at least 3 or one for which there exists a subgraph with some characteristic always contains a proper 3-dominating set. Further, we find that when certain conditions are met, some graph products, such as the joint product, strong product, lexicographic product, and corona product of two graphs, have a proper 3-dominating set. Moreover, we discover the bounds of the proper 3-domination number. For some special graphs, we get their proper 3-domination numbers.