Disjoint dominating and total dominating sets in graphs

Abstract

a b s t r a c tIt has been shown [M.A. Henning, J. Southey, A note on graphs with disjoint dominating and total dominating sets, Ars Combin. 89 (2008) 159-162] that every connected graph with minimum degree at least two that is not a cycle on five vertices has a dominating set D and a total dominating set T which are disjoint. We characterize such graphs for which D ∪ T necessarily contains all vertices of the graph and that have no induced cycle on five vertices.

Thoroughly dispersed colorings

“…We will also need the result that every graph has two disjoint sets, one total dominating and one “half‐total” dominating: Theorem () If G is a graph with minimum degree at least , then, except for the 5‐cycle, G has disjoint dominating and total dominating sets.…”

“…Theorem 15. ( [15]) If is a graph with minimum degree at least , then, except for the 5-cycle, has disjoint dominating and total dominating sets.…”

Partitioning a graph into a dominating set, a total dominating set, and something else

“…In contrast to that, Zelinka [14,15] observed that no minimum degree condition is sufficient for the existence of three disjoint dominating sets or of two disjoint total dominating sets. In [9] Henning and Southey proved the following result which is somehow located between Ore's positive and Zelinka's negative observation.…”

“…Theorem 1 (Henning and Southey [9]) If G is a graph of minimum degree at least 2 such that no component of G is a chordless cycle of length 5, then V (G) can be partitioned into a dominating set D and a total dominating set T .…”

A theorem of Ore and self-stabilizing algorithms for disjoint minimal dominating sets

“…Among these is a paper by Henning and Southey [16] in which they prove that any connected graph having minimum degree at least two, and not a cycle on 5 vertices, possesses a dominating set and a disjoint total dominating set (a dominating set S for which every node in V , rather than V − S, has a neighbor in S). It would be interesting to see if the algorithms in this paper can be modified to find disjoint dominating sets of different types.…”