Total domination in partitioned trees and partitioned graphs with minimum degree two

Abstract

Let G = (V, E) be a graph and let $$S \subseteq V$$ . A set of vertices in G totally dominates S if every vertex in S is adjacent to some vertex of that set. The least number of vertices needed in G to totally dominate S is denoted by ? t (G, S). When S = V, ? t (G, V) is the well studied total domination number ? t (G). We wish to maximize the sum ? t (G) + ? t (G, V 1) + ? t (G, V 2) over all possible partitions V 1, V 2 of V. We call this maximum sum f t (G). For a graph H, we denote by H ^ P 2 the graph obtained from H by attaching a path of length 2 to each vertex of H so that the resulting paths are vertex-disjoint. We show that if G is a tree of order n ? 4 and $$G \notin \{P_5, P_6, P_7, P_{10}, P_{14}\}$$ , then f t (G) ? 14n/9 with equality if and only if G ?{P 9, P 18} or G = (T ^ P 2) ^ P 2 for some tree T. If G is a connected graph of order n with minimum degree at least two, we establish that f t (G) ? 3n/2 with equality if and only if G is a cycle of order congruent to zero modulo 4.