Let G be an undirected connected graph with vertex and edge sets V (G) E(G), respectively. A set C ⊆ is called weakly convex hop dominating if for every two vertices x, y ∈ C, there exists x-y geodesic P(x, y) such that (P(x, y)) v (G)\C, w dG(v, w) = 2. The minimum cardinality of a G, denoted by γwconh(G), the domination number G. In this paper, we introduce initially investigate concept domin...

A subset S of V (G), where G is a simple undirected graph, hop dominating if for each v ∈ (G) \ S, there exists w such that dG(v, w) = 2 and it differentiating N2 G[u] ∩ ̸= G[v] any two distinct vertices u, (G). set ⊆ both in G. The minimum cardinality G, denoted by γdh(G), called the domination number In this paper, we investigate some properties newly defined parameter. particular, characteriz...

Abstract In this paper, we study the problem of deciding whether total domination number a given graph G can be reduced using exactly one edge contraction (called 1 -Edge Contraction( ? t ) ). We focus on several classes and determine computational complexity problem. By putting together these results, manage to obtain complete dichotomy for H-free graphs.

We prove that the problem of nding two minimum dominating sets (connected dominating sets or vertex clique covers) with minimum intersection is linearly solvable in interval graphs. Furthermore, the problem of deciding whether or not there exist two disjoint minimum dominating sets (connected dominating sets or vertex clique covers) is shown to be NP-hard for chordal graphs.