#### Title

#### Document Type

Article

#### Publication Date

3-17-2013

#### Publication Title

Graphs and Combinatorics

#### Abstract

Abstract. Given a graph G, the k-dominating graph of G, D_{k}(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D_{k}(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D_{k}(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D_{k}(G). In this paper we give conditions that ensure D_{k}(G) is connected.

#### Keywords

Domination, Reconfiguration problems, Chordal graphs, Bipartite graphs

#### Volume

30

#### Issue

3

#### First Page

609

#### Last Page

617

#### DOI

10.1007/s00373-013-1302-3

#### ISSN

1435-5914

#### Recommended Citation

Haas, Ruth and Seyffarth, Karen, "The k-Dominating Graph" (2013). *Mathematics and Statistics: Faculty Publications*. 3.

http://scholarworks.smith.edu/mth_facpubs/3

## Comments

Peer reviewed post-print.

The final publication is available at Springer via http://dx.doi.org/10.1007/s00373-013-1302-3