On total domination vertex critical graphs of high connectivity

AbstractA graph is called γ-critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γt-critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γt-critical graph that has total domination number k, is called k-γt-critical. In this paper, we introduce a class of k-γt-critical graphs of high connectivity for each integer k≥3. In particular, we provide a partial answer to the question “Which graphs are γ-critical and γt-critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255–261]

Connected majority domination vertex critical graphs

In this article, how the removal of a single vertex from a graph G can change the Connected Majority Domination number is surveyed for any graph G. A graph is Connected Domination Critical if the removal of any vertex decreases or increases its Connected Majority Domination Number. This paper gives examples and properties of CMD vertex critical graphs. There are two types namely CVR and UVR with respect to CMD sets of a graph. Also the vertex classification. V⁰CM(G), V ⁻CM(G) and V⁺CM are studied, characterisation theorems of these vertex classification are determined

A constructive characterization of total domination vertex critical graphs

AbstractLet G be a graph of order n and maximum degree Δ(G) and let γt(G) denote the minimum cardinality of a total dominating set of a graph G. A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G−v is less than the total domination number of G. We call these graphs γt-critical. For any γt-critical graph G, it can be shown that n≤Δ(G)(γt(G)−1)+1. In this paper, we prove that: Let G be a connected γt-critical graph of order n (n≥3), then n=Δ(G)(γt(G)−1)+1 if and only if G is regular and, for each v∈V(G), there is an A⊆V(G)−{v} such that N(v)∩A=0̸, the subgraph induced by A is 1-regular, and every vertex in V(G)−A−{v} has exactly one neighbor in A

Domination changing and unchanging signed graphs upon the vertex removal

A subset S of V (Σ) is a dominating set of Σ if |N⁺(v) ∩ S| > |N⁻(v) ∩ S| for all v ∈ V − S. This article is to start a study of those signed graphs that are stable and critical in the following way: If the removal of an arbitrary vertex does not change the domination number, the signed graph will be stable. The signed graph, on the other hand, is unstable if an arbitrary vertex is removed and the domination number changes. Specifically, we analyze the change in the domination of the vertex deletion and stable signed graphs.Publisher's Versio