Let (G) be the domination number of graph G, thus a graph G is k -edge-critical if (G) 1⁄4 k ; and for every nonadjacent pair of vertices u and v, (Gþ uv) 1⁄4 k 1. In Chapter 16 of the book ‘‘Domination in Graphs— Advanced Topics,’’ D. Sumner cites a conjecture of E. Wojcicka under the form ‘‘3-connected 4-critical graphs are Hamiltonian and perhaps, in general (i.e., for any k 4), (k 1)-connec...

‎Let $G=(V,E)$ be a graph‎. ‎A subset $Ssubset V$ is a hop dominating set‎‎if every vertex outside $S$ is at distance two from a vertex of‎‎$S$‎. ‎A hop dominating set $S$ which induces a connected subgraph‎ ‎is called a connected hop dominating set of $G$‎. ‎The‎‎connected hop domination number of $G$‎, ‎$ gamma_{ch}(G)$,‎‎‎ ‎is the minimum cardinality of a connected hop‎‎dominating set of $G$...

A dominating set D 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 set C 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 set W of a g...

A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(...

For an integer k ≥ 1, a (distance) k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V (G) \ S is at distance at most k from some vertex of S. The k-domination number, γk(G), of G is the minimum cardinality of a k-dominating set of G. In this talk, we establish lower bounds on the k-domination number of a graph in terms of its diameter, radius and gir...

If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number γc(G) of G. A graph G is called a perfect connected-dominant graph if γ(H) = γc(H) for each connected induced subgraph H of G. We prove that a graph is a perfect connected-dominant graph if and only i...