Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6871558 | Discrete Applied Mathematics | 2018 | 10 Pages |
Abstract
A set D of vertices in an isolate-free graph G is a total dominating set of G if every vertex is adjacent to a vertex in D. The total domination number, γt(G), of G is the minimum cardinality of a total dominating set of G. We note that γt(G)â¥2 for every isolate-free graph G. A non-isolating set of vertices in G is a set of vertices whose removal from G produces an isolate-free graph. The γtâ-stability, denoted stγtâ(G), of G is the minimum size of a non-isolating set S of vertices in G whose removal decreases the total domination number. We show that if G is a connected graph with maximum degree Πsatisfying γt(G)â¥3, then stγtâ(G)â¤2Îâ1, and we characterize the infinite family of trees that achieve equality in this upper bound. The total domination stability, stγt(G), of G is the minimum size of a non-isolating set of vertices in G whose removal changes the total domination number. We prove that if G is a connected graph with maximum degree Î satisfying γt(G)â¥3, then stγt(G)â¤2Îâ1.
Keywords
Related Topics
Physical Sciences and Engineering
Computer Science
Computational Theory and Mathematics
Authors
Michael A. Henning, Marcin Krzywkowski,