Graphs with isolation number equal to one third of the order

A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is known that $\iota (G)\le n/3$, if $G$ is a connected graph of order $n$, $n\ge 3$, distinct from $C_5$. The main result of this work is the characterisation of unicyclic and block graphs of order $n$ with isolating number equal to $n/3$. Moreover, we provide a family of general graphs attaining this upper bound on the isolation number.Comment: 15 pages, 12 figure

Isolation Number versus Domination Number of Trees

[Abstract] If = (Vɢ,Eɢ) is a graph of order n, we call ⊆ Vɢ an isolating set if the graph induced by Vɢ − Nɢ[] contains no edges. The minimum cardinality of an isolating set of is called the isolation number of , and it is denoted by (). It is known that () ≤ ⁿ⁄₃ and the bound is sharp. A subset ⊆ Vɢ is called dominating in if Nɢ[] = Vɢ. The minimum cardinality of a dominating set of is the domination number, and it is denoted by (). In this paper, we analyze a family of trees where () = (), and we prove that (T) = ⁿ⁄₃ implies () = (). Moreover, we give different equivalent characterizations of such graphs and we propose simple algorithms to build these trees from the connections of stars.CITIC, as Research Center accredited by Galician University System, is funded by "Consellería de Cultura, Educación e Universidade from Xunta de Galicia", supported in an 80% through ERDF Funds, ERDF Operational Programme Galicia 2014-2020, and the remaining 20% by "Secretaría Xeral de Universidades (Grant ED431G 2019/01). This research was also funded by Agencia Estatal de Investigación of Spain (PID2019-104958RB-C42 and TIN2017-85160-C2-1-R) and ERDF funds of the EU (AEI/FEDER, UE).Xunta de Galicia; ED431G 2019/0

Similarities and Differences Between the Vertex Cover Number and the Weakly Connected Domination Number of a Graph

peer reviewe

Nordhaus-Gaddum results for the convex domination number of a graph

10.1007/s10998-012-2174-7The distance d G (u, v) between two vertices u and v in a connected graph G is the length of the shortest uv-path in G. A uv-path of length d G (u, v) is called a uv-geodesic. A set X is convex in G if vertices from all ab-geodesics belong to X for any two vertices a, b ? X. The convex domination number ?con(G) of a graph G equals the minimum cardinality of a convex dominating set. In the paper, Nordhaus-Gaddum-type results for the convex domination number are studied