Geodesic
Outer connected domination in maximal outerplanar graphs and beyond
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 575-590

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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 γ̃_̃c̃(G), 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 γ̃_̃c̃(G)≤⌊n+k4⌋ 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 γ̃_̃c̃(G)=⌊n+24⌋ 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 K_2,3-minor free graphs without K_4 and all K_4-minor free graphs.
Keywords: maximal outerplanar graphs, outer connected domination, striped maximal outerplanar graphs 
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     title = {Outer connected domination in maximal outerplanar graphs and beyond},
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Yang, Wei; Wu, Baoyindureng. Outer connected domination in maximal outerplanar graphs and beyond. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 575-590. http://geodesic.mathdoc.fr/item/DMGT_2024_44_2_a8/