a b s t r a c tIn this paper we introduce the notion of distance k-guarding applied to triangulation graphs, and associate it with distance k-domination and distance k-covering. We obtain results for maximal outerplanar graphs when k = 2. A set S of vertices in a triangulation graph T is a distance 2-guarding set (or 2d-guarding set for short) if every face of T has a vertex adjacent to a vertex of S. We show that ⌊ n 5 ⌋ (respectively, ⌊ n 4 ⌋) vertices are sufficient to 2d-guard and 2d-dominate (respectively, 2d-cover) any n-vertex maximal outerplanar graph. We also show that these bounds are tight.

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.

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