The strong metric dimension of graphs and digraphs

Let G be a connected (di)graph. A vertex w is said to strongly resolve a pair u,v of vertices of G if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong dimension of G. It is shown that the problem of finding the strong dimension of a connected graph can be transformed to the problem of finding the vertex covering number of a graph. Moreover, it is shown that computing this invariant is NP-hard. Related invariants for directed graphs are defined and studied.     

Connected domination of regular graphs

A dominating setD of a graph G is a subset of V(G) such that for every vertex v∈V(G), either v∈D or there exists a vertex u∈D that is adjacent to v in G. Dominating sets of small cardinality are of interest. A connected dominating setC of a graph G is a dominating set of G such that the subgraph induced by the vertices of C in G is connected. A weakly-connected dominating setW of a graph G is a dominating set of G such that the subgraph consisting of V(G) and all edges incident with vertices in W is connected. In this paper we present several algorithms for finding small connected dominating sets and small weakly-connected dominating sets of regular graphs. We analyse the average-case performance of these heuristics on random regular graphs using differential equations, thus giving upper bounds on the size of a smallest connected dominating set and the size of a smallest weakly-connected dominating set of random regular graphs.     

More rotation numbers for complete bipartite graphs

Let G be a simple undirected graph which has p vertices and is rooted at x. Informally, the rotation number h(G, x) of this rooted graph is the minimum number of edges in a p vertex graph H such that for each vertex v of H, there exists a copy of G in H with the root x at v. In this article we calculate some rotation numbers for complete bipartite graphs, and thus greatly extend earlier results of Cockayne and Lorimer.     

Minimal extensions of graphs to absolute retracts

A graph H is an absolute retract if for every isometric embedding h of, into a graph G an edge-preserving map g from G to H exists such that g · h is the identity map on H. A vertex v is embeddable in a graph G if G ? v is a retract of G. An absolute retract is uniquely determined by its set of embeddable vertices. We may regard this set as a metric space. We also prove that a graph (finite metric space with integral distance) can be isometrically embedded into only one smallest absolute retract (injective hull). All graphs in this paper are finite, connected, and without multiple edges.     

A survey of stratified domination in graphs

A graph G is 2-stratified if its vertex set is partitioned into two nonempty classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F in G rooted at v. In this paper, we survey recent results on the F-domination number for various 2-stratified graphs F.     

Eccentric sequences and eccentric sets in graphs

By a graph we mean a finite undirected connected graph of order p, p ? 2, with no loops or multiple edges. A finite non-decreasing sequence S: s₁, s₂, …, s_p, p ? 2, of positive integers is an eccentric sequence if there exists a graph G with vertex set V(G) = {v₁, v₂, …, v_p} such that for each i, 1 ? i ? p, s, is the eccentricity of v₁. A set S is an eccentric set if there exists a graph G such that the eccentricity ρ(v₁) is in S for every v₁ ? V(G), and every element of S is the eccentricity of some vertex in G. In this note we characterize eccentric sets, and we find the minimum order among all graphs whose eccentric set is a given set, to obtain a new necessary condition for a sequence to be eccentric. We also present some properties of graphs having preassigned eccentric sequences.     

Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

On geodetic sets formed by boundary vertices

Let G be a finite simple connected graph. A vertex v is a boundary vertex of G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices. Before showing that one of the main results in [G. Chartrand, D. Erwin, G.L. Johns, P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25-34] does not hold for one of the cases, we establish a realization theorem that not only corrects the mentioned wrong statement but also improves it.Given S⊆V(G), its geodetic closure I[S] is the set of all vertices lying on some shortest path joining two vertices of S. We prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, I[∂(G)]=V(G). A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than v. We present some sufficient conditions to guarantee the geodeticity of either the contour Ct(G) or its geodetic closure I[Ct(G)].     

Special Eccentric Vertices for the Class of Chordal Graphs and Related Classes

A vertex is simplicial if the vertices of its neighborhood are pairwise adjacent. It is known that, for every vertex v of a chordal graph, there exists a simplicial vertex among the vertices at maximum distance from v. Here we prove similar properties in other classes of graphs related to that of chordal graphs. Those properties will not be in terms of simplicial vertices, but in terms of other types of vertices that are used to characterize those classes.     

Which trees are link graphs?

The link of a vertex v of a graph G is the subgraph induced by all vertices adjacent to v. If all the links of G are isomorphic to L, then G has constant link and L is called a link graph. We investigate the trees of order p≤9 to see which are link graphs. Group theoretic methods are used to obtain constructions of graphs G with constant link L for certain trees L. Necessary conditions are derived for the existence of a graph having a given graph L as its constant link. These conditions show that many trees are not link graphs. An example is given to show that a connected graph with constant link need not be point symmetric.     

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.     

On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.     

A Generalization of Opsut’s Lower Bounds for the Competition Number of a Graph

The notion of a competition graph was introduced by Cohen in 1968. The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. In 1978, Roberts defined the competition number k(G) of a graph G as the minimum number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. In 1982, Opsut gave two lower bounds for the competition number of a graph. In this paper, we give a generalization of these two lower bounds for the competition number of a graph.     

Graph Equation for Line Graphs and m-Step Graphs

Given a graph G, the m-step graph of G, denoted by S _m (G), has the same vertex set as G and an edge between two distinct vertices u and v if there is a walk of length m from u to v. The line graph of G, denoted by L(G), is a graph such that the vertex set of L(G) is the edge set of G and two vertices u and v of L(G) are adjacent if the edges corresponding to u and v share a common end vertex in G. We characterize connected graphs G such that S _m (G) and L(G) are isomorphic.     

Graphs with a Minimal Number of Convex Sets

Let G be a graph with vertex set V(G). A set C of vertices of G is g-convex if for every pair \({u, v \in C}\) the vertices on every u–v geodesic (i.e. shortest u–v path) belong to C. If the only g-convex sets of G are the empty set, V(G), all singletons and all edges, then G is called a g-minimal graph. It is shown that a graph is g-minimal if and only if it is triangle-free and if it has the property that the convex hull of every pair of non-adjacent vertices is V(G). Several properties of g-minimal graphs are established and it is shown that every triangle-free graph is an induced subgraph of a g-minimal graph. Recursive constructions of g-minimal graphs are described and bounds for the number of edges in these graphs are given. It is shown that the roots of the generating polynomials of the number of g-convex sets of each size of a g-minimal graphs are bounded, in contrast to their behaviour over all graphs. A set C of vertices of a graph is m-convex if for every pair \({u, v \in C}\) , the vertices of every induced u–v path belong to C. A graph is m-minimal if it has no m-convex sets other than the empty set, the singletons, the edges and the entire vertex set. Sharp bounds on the number of edges in these graphs are given and graphs that are m-minimal are shown to be precisely the 2-connected, triangle-free graphs for which no pair of adjacent vertices forms a vertex cut-set.     

On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.     

The competition number of a graph and the dimension of its hole space

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph has been studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is not smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.     

Characterizations of 2-variegated graphs and of 3-variegated graphs

A graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. We prove that a graph G on 2n vertices is 2-variegated if and only if there exists a set S of n independent edges in G such that no cycle in G contains an odd number of edges from S. We also characterize 3-variegated graphs.     

Geodeticity of the contour of chordal graphs

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.