Induced Subgraphs of Gamma Graphs

Abstract

Let G be a graph. The gamma graph of G denoted by γ ⋅ G is the graph with vertex set V(γ ⋅ G) as the set of all γ-sets of G and two vertices D and S of γ ⋅ G are adjacent if and only if |D ∩ S| = γ(G) – 1. A graph H is said to be a γ-graph if there exists a graph G such that γ ⋅ G is isomorphic to H. In this paper, we show that every induced subgraph of a γ-graph is also a γ-graph. Furthermore, if we prove that H is a γ-graph, then there exists a sequence {Gn} of non-isomorphic graphs such that H = γ ⋅ Gn for every n.

Introduction to Reconfiguration

“…The graph γ · G [116] uses k = γ(G) under TJ, whereas the γ-graph of G uses k = γ(G) under TS [117], later considered for different variants such as the total dominating number, paired domination number, and connected dominating number [118]. Work on both the TJ model [119][120][121][122][123][124][125] and the TS model [126] has focused on characterizing which graphs are gamma graphs under the various definitions.…”

Introduction to Reconfiguration

“…The graph γ · G [116] uses k = γ(G) under TJ, whereas the γ-graph of G uses k = γ(G) under TS [117], later considered for different variants such as the total dominating number, paired domination number, and connected dominating number [118]. Work on both the TJ model [119][120][121][122][123][124] and the TS model [125] has focused on characterizing which graphs are gamma graphs under the various definitions.…”

Reinforcement Number of a Graph with respect to Half-Domination

“…e total domination number of a graph G, denoted by c t (G), is the minimum cardinality of a total dominating set of G and, for their properties, we refer the reader to [2][3][4][5][6]. In [7], Kulli introduced the concept of the cobondage number cb(G) of a graph, which is the minimum number of edges to be added to reduce the domination number.…”