The domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [9] that sd(T ) ≤ 3 for any tree T . We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number of a nonempty graph G as a minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. We show that msd(G) ≤ 3 for any graph G. The domination subdivision number and the domination multisubdivision numer of a graph are incomparable in general case, but we show that for trees these two parameters are equal. We also determine domination multisubdivision number for some classes of graphs.
The neighbourhood of a vertex v of a graph G is the set N (v) of all verticesThe super domination number of G is the minimum cardinality among all super dominating sets in G. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard.
Imagine that we are given a set D of officials and a set W of civils. For each civil x ∈ W , there must be an official v ∈ D that can serve x, and whenever any such v is serving x, there must also be another civil w ∈ W that observes v, that is, w may act as a kind of witness, to avoid any abuse from v. What is the minimum number of officials to guarantee such a service, assuming a given social network?In this paper, we introduce the concept of certified domination that perfectly models the aforementioned problem. Specifically, a dominating set D of a graph G = (V G , E G ) is said to be certified if every vertex in D has either zero or at least two neighbours in V G \ D. The cardinality of a minimum certified dominating set in G is called the certified domination number of G. Herein, we present the exact values of the certified domination number for some classes of graphs as well as provide some upper bounds on this parameter for arbitrary graphs. We then characterise a wide class of graphs with equal domination and certified domination numbers and characterise graphs with large values of certified domination numbers. Next, we examine the effects on the certified domination number when the graph is modified by deleting/adding an edge or a vertex. We also provide Nordhaus-Gaddum type inequalities for the certified domination number. Finally, we show that the (decision) certified domination problem is NP-complete. As a side result, we characterise a wider class of DD 2 -graphs, thus generalizing a result of [19].
In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G • P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.
A set X is weakly convex in G if for any two vertices a, b ∈ X there exists an ab-geodesic such that all of its vertices belong to X. A set X ⊆ V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number γ wcon (G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd γwcon (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and establish upper bounds for it.
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