On the Structure of Graphs Vertex Critical with Respect to Connected Domination

Michael D. Plummer ()
Additional contact information
Michael D. Plummer: Vanderbilt University, Department of Mathematics

Chapter 14 in Research Trends in Combinatorial Optimization, 2009, pp 303-315 from Springer

Abstract: Summary A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γ c (G) denote the size of any smallest connected dominating set in G. Graph G is k-γ-connected-vertex-critical (abbreviated “kcvc”) if $\gamma_{c}(G)=\nobreak k$ , but if any vertex v is deleted from G, then γ c (G−v)≤k−1. This concept of vertex criticality stands in contrast to the concept of criticality with respect to edge addition in which a graph G is defined to be k-connected-critical if the connected domination number of G is k, but if any edge is added to G, the connected domination number falls to k−1. It is well-known that the only 1cvc graph is K 1 and the 2cvc graphs are obtained from the even complete graphs K 2n , with n≥2, by deleting a perfect matching. In this paper we survey some recent results for the case when γ c =3. In Sect. 14.2 we present some recently derived basic properties of 3cvc graphs, especially with respect to connectivity, and then present three new infinite families of 3cvc graphs. In Sect. 14.3, we present some new matching results for 3cvc graphs.

Date: 2009
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-76796-1_14

Ordering information: This item can be ordered from
http://www.springer.com/9783540767961

DOI: 10.1007/978-3-540-76796-1_14

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().