Connected domination in maximal outerplanar graphs

Zhuang, Wei

doi:10.1016/j.dam.2020.01.033

Cited by 3 publications

(1 citation statement)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More details and information can be found in [1,2,11,13,15]. Various types of domination of maximal outerplanar graphs were widely studied in literature, such as secure domination [3], partial domination [5], domination [6,12,14], total domination [8,9], semipaired domination [10] and connected domination [18]. Very recently, Zhuang [17] proved the following theorem.…”

Section: Introductionmentioning

confidence: 99%

Outer connected domination in maximal outerplanar graphs and beyond

Yang¹,

Wu²

2024

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

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 γc (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 γc (G) ≤ n+k 4 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 γc (G) = n+2 4 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.

show abstract