Outer Connected Domination in Maximal Outerplanar Graphs and Beyond

Abstract

Abstract A set S of vertices in a graph G is an outer connected dominating set of G if every vertex in V \ S is adjacent to a vertex in S and the subgraph induced by V \ S is connected. The outer connected domination number of G, denoted by γ˜c(G) {\tilde \gamma _c}\left( G \right) , is the minimum cardinality of an outer connected dominating set of G. Zhuang [Domination and outer connected domination in maximal outerplanar graphs, Graphs Combin. 37 (2021) 2679–2696] recently proved that γ˜c(G)≤⌊ n+k4 ⌋ {\tilde \gamma _c}\left( G \right) \le \left\lfloor {{{n + k} \over 4}} \right\rfloor for any maximal outerplanar graph G of order n ≥ 3 with k vertices of degree 2 and posed a conjecture which states that G is a striped maximal outerplanar graph with γ˜c(G)≤⌊ n+24 ⌋ {\tilde \gamma _c}\left( G \right) \le \left\lfloor {{{n + 2} \over 4}} \right\rfloor if and only if G ∈ 𝒜, where 𝒜 consists of six special families of striped outerplanar graphs. We disprove the conjecture. Moreover, we show that the conjecture become valid under some additional property to the striped maximal outerplanar graphs. In addition, we extend the above theorem of Zhuang to all maximal K2,3-minor free graphs without K4 and all K4-minor free graphs.