Graphs & digraphs (2nd ed.)
Graphs with large total domination number
Journal of Graph Theory
| Hi-index | 0.03 |
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 驴 t (G, S). When S = V, 驴 t (G, V) is the well studied total domination number 驴 t (G). We wish to maximize the sum 驴 t (G) + 驴 t (G, V 1) + 驴 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 驴 4 and $$G \notin \{P_5, P_6, P_7, P_{10}, P_{14}\}$$ , then f t (G) 驴 14n/9 with equality if and only if G 驴{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) 驴 3n/2 with equality if and only if G is a cycle of order congruent to zero modulo 4.