Combinatorial bounds on connectivity for dominating sets in maximal outerplanar graphs

Abstract

A b stractIn this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs. We establish tight combinatorial bounds for con nected domination, semitotal domination, independent domination and weakly con nected domination for any n-vertex maximal outerplaner graph.

Total domination in plane triangulations

“…, j, we obtain a TDS of size at most f (n − 4j − 1) + 1 + 1 + 3 2 (j − 1). By Lemma 12, f (n − 4j − 1) + 1 + 1 + 3 2 (j − 1) ≤ f (n) because 2+3(j−1)/2 4j+1 ≤ 2/5, hence γ t (T ) ≤ f (n). Note that v j is dominated by u j .…”

“…If d k is (3,6) or (4,6), then it defines a MOP of size at least 10 and we are in Case 6. If d k is (3,7), then it defines a MOP of size 9, where (3, 7) would be contractible in M k and we would be in Case 5. Hence, M is neither H 1 nor H 2 .…”

“…In [17], Lemanska et al presented an alternative proof of this last result. The reader is referred to [3,2,4,12,16] for other results in MOPs related to some variants of the domination concept.…”

“…P is terminal, hence some vertices of H 1 must be interior vertices in M , implying that d k is a diagonal of H 1 . Thus, by the symmetry of H 1 (see Figure 1), d k can only be one of the edges (3, 7), (3,6) and (4,6). If d k is (3,6) or (4,6), then it defines a MOP of size at least 10 and we are in Case 6.…”

A generalization of the Art Gallery Theorem

“…Fisk [8] and Matheson and Tarjan [12] also gave alternative proofs. For results on other types of domination in mops, we refer the reader to [3,5,6,7,9,15]. Caro and Hansberg [2] proved that the K 1,1isolation number of a mop of order n ≥ 4 is at most n/4.…”