Elsevier

Discrete Applied Mathematics

Volume 198, 10 January 2016, Pages 164-169
Discrete Applied Mathematics

On dominating sets of maximal outerplanar and planar graphs

https://doi.org/10.1016/j.dam.2015.06.024Get rights and content
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Abstract

A set DV(G) of a graph G is a dominating set if every vertex vV(G) is either in D or adjacent to a vertex in D. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G. Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n-vertex maximal outerplanar graph having t vertices of degree 2, then γ(G)n+t4. We improve their upper bound by showing γ(G)n+k4, where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ(G)5n16 for a Hamiltonian maximal planar graph G with n7 vertices.

Keywords

Dominating set
Domination number
Maximal planar graph
Maximal outerplanar graph

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Supported by 973 Program of China 2013CB329600, National Natural Science Foundation of China Grant 61309015 and National Natural Science Foundation of China Special Equipment Grant 61127005.