Weakly connected domination critical graphs

• Main Authors: Magdalena Lemańska, Agnieszka Patyk
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2008-01-01
• Series: Opuscula Mathematica
• Subjects: weakly connected domination number; tree; critical graphs
• Online Access: http://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2824.pdf

A dominating set \(D \subset V(G)\) is a weakly connected dominating set in \(G\) if the subgraph \(G[D]_w = (N_{G}[D],E_w)\) weakly induced by \(D\) is connected, where \(E_w\) is the set of all edges with at least one vertex in \(D\). The weakly connected domination number \(\gamma_w(G)\) of a graph \(G\) is the minimum cardinality among all weakly connected dominating sets in \(G\). The graph is said to be weakly connected domination critical (\(\gamma_w\)-critical) if for each \(u, v \in V(G)\) with \(v\) not adjacent to \(u\), \(\gamma_w(G + vu) \lt \gamma_w (G)\). Further, \(G\) is \(k\)-\(\gamma_w\)-critical if \(\gamma_w(G) = k\) and for each edge \(e \not\in E(G)\), \(\gamma_w(G + e) \lt k\). In this paper we consider weakly connected domination critical graphs and give some properties of \(3\)-\(\gamma_w\)-critical graphs.