Let G be a connected graph. Given an ordered set W = {w1, . . . , wk} ⊆ V (G) and a vertex u ∈ V (G), the representation of u with respect to W is the ordered k-tuple (d(u, w1), d(u, w2), . . . , d(u, wk)), where d(u, wi) denotes the distance between u and wi. The set W is a metric generator for G if every two different vertices of G have distinct representations. A minimum cardinality metric generator is called a metric basis of G and its cardinality is called the metric dimension of G. It is well...

A β-perfect graph is a simple graph G such that χ(G') = β(G') for every induced subgraph G' of G, where χ(G') is the chromatic number of G', and β(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs,...

A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The connected domatic number of $G$ is the maximum number of pairwise disjoint, connected dominating sets in $V\left(G\right)$. We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.

A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The minimum number of vertices in a connected dominating set of $G$ is called the connected domination number of $G$, and is denoted by ${\gamma }_c\left(G\right)$. Let $G$ be a spanning subgraph of $K_{s,s}$ and let $H$ be the complement of $G$ relative to $K_{s,s}$; that is, $K_{s,s}=G\oplus H$ is a factorization of $K_{s,s}$. The graph $G$ is $k$-${\gamma }_c$-critical relative to $K_{s,s}$ if ${\gamma }_c\left(G\right)=k$ and ${\gamma }_c(G+e)<k$ for each edge $e\in E\left(H\right)$. First, we discuss some classes of graphs whether they are ${\gamma }_c$-critical relative...

We consider finite graphs G with vertex set V(G). For a subset S ⊆ V(G), we define by G[S] the subgraph induced by S. By n(G) = |V(G) | and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V(G) is a connected global offensive k-alliance of the connected graph G, if G[S] is connected and |N(v) ∩ S | ≥ |N(v) -S | + k for every vertex v ∈ V(G) -S, where N(v) is the neighborhood of v. The connected global offensive k-alliance number $\gamma {ₒ}^{k,c}\left(G\right)$ is the...

An odd dominating set of a simple, undirected graph G = (V,E) is a set of vertices D ⊆ V such that |N[v] ∩ D| ≡ 1 mod 2 for all vertices v ∈ V. It is known that every graph has an odd dominating set. In this paper we consider the concept of connected odd dominating sets. We prove that the problem of deciding if a graph has a connected odd dominating set is NP-complete. We also determine the existence or non-existence of such sets in several classes of graphs. Among other results, we prove there...

For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = mind(v,x)|x ∈ S. For an ordered k-partition Π = S₁,S₂,...,Sₖ of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S₁), d(v,S₂),..., d(v,Sₖ)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = S₁,S₂,...,Sₖ...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r\left(v\right|W)$ = ($d(v,w_1)$, $d(v,w_2),\cdots ,d(v,w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $dim\left(G\right)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph...