Domination in graphs of minimum degree at least two and large girth

We prove that for graphs of order n, minimum degree 2 and girth g 5 the domination number satisfies 1 3 + 2 3gn. As a corollary this implies that for cubic graphs of order n and girth g 5 the domination number satisfies 44 135 + 82 135gn which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic graphs, manuscript (2005)) and Kawarabayashi, Plummer and Saito (Domination in a graph with a 2-factor, J. Graph Theory 52 (2006), 1-6) for large enough girth. Furthermore, it confirms a conjecture due to Reed about connected cubic graphs (Paths, stars and the number three, Combin. Prob. Comput. 5 (1996), 267-276) for girth at least 83

Improved upper bounds on the domination number of graphs with minimum degree at least five

An algorithmic upper bound on the domination number $\gamma$ of graphs in terms of the order $n$ and the minimum degree $\delta$ is proved. It is demonstrated that the bound improves best previous bounds for any $5\le \delta \le 50$. In particular, for $\delta=5$, Xing et al.\ proved in 2006 that $\gamma \le 5n/14 < 0.3572 n$. This bound is improved to $0.3440 n$. For $\delta=6$, Clark et al.\ in 1998 established $\gamma <0.3377 n$, while Bir\'o et al. recently improved it to $\gamma <0.3340 n$. Here the bound is further improved to $\gamma < 0.3159 n$. For $\delta=7$, the best earlier bound $0.3 088 n$ is improved to $\gamma < 0.2927 n$

Domination in bipartite graphs

We prove that the domination number of a graph of order n and minimum degree at least 2 that does not contain cycles of lengths 4, 5, 7, 10 or 13 is at most 3 8n. Furthermore, we derive upper bounds on the domination number of bipartite graphs of given minimum degre

In the complement of a dominating set

A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex of D\V has at least one neighbor that belongs to D. The disjoint domination number of a graph G is the minimum cardinality of two disjoint dominating sets of G. We prove upper bounds for the disjoint domination number for graphs of minimum degree at least 2, for graphs of large minimum degree and for cubic graphs.A set T of vertices of a graph G=(V,E) is a total dominating set, if every vertex of G has at least one neighbor that belongs to T. We characterize graphs of minimum degree 2 without induced 5-cycles and graphs of minimum degree at least 3 that have a dominating set, a total dominating set, and a non-empty vertex set that are disjoint.A set I of vertices of a graph G=(V,E) is an independent set, if all vertices in I are not adjacent in G. We give a constructive characterization of trees that have a maximum independent set and a minimum dominating set that are disjoint and we show that the corresponding decision problem is NP-hard for general graphs. Additionally, we prove several structural and hardness results concerning pairs of disjoint sets in graphs which are dominating, independent, or both. Furthermore, we prove lower bounds for the maximum cardinality of an independent set of graphs with specifed odd girth and small average degree.A connected graph G has spanning tree congestion at most s, if G has a spanning tree T such that for every edge e of T the edge cut defined in G by the vertex sets of the two components of T-e contains at most s edges. We prove that every connected graph of order n has spanning tree congestion at most n^(3/2) and we show that the corresponding decision problem is NP-hard