Cohabitation of independent sets and dominating sets in trees

We give a constructive characterization of trees that have a maximum independent set and a minimum dominating set which are disjoint and show that the corresponding decision problem is NP-complete for general graphs

Pairs of disjoint dominating sets and the minimum degree of graphs

For a connected graph G of order n and minimum degree \delta we prove the existence of two disjoint dominating sets D_1 and D_2 such that, if \delta \geq 2, then \mid D_1 \cup D_2 \mid \leq \frac{6}{7} n unless G = C_4, and, if \delta \geq 5, then \mid D_1 \cup D_2 \mid \leq 2 \frac{1+ln(\delta +1)}{\delta + 1}n. While for the first estimate there are exactly six extremal graphs which are all of order 7, the second estimate is asymptotically best-possible

Pairs of disjoint dominating sets and the minimum degree of graphs

For a connected graph G of order n and minimum degree \delta we prove the existence of two disjoint dominating sets D_1 and D_2 such that, if \delta\geq 2, then |D_1\cup D_2|\leq \frac{6}{7}n unless G=C_4, and, if \delta\geq 5, then |D_1\cup D_2|\leq 2\frac{1+\ln(\delta+1)}{\delta+1}n. While for the first estimate there are exactly six extremal graphs which are all of order 7, the second estimate is asymptotically best-possible

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