Locating-dominating sets in twin-free graphs

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $\gamma_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] that if $G$ is a twin-free graph of order $n$ without isolated vertices, then $\gamma_L(G)\le \frac{n}{2}$. We prove the general bound $\gamma_L(G)\le \frac{2n}{3}$, slightly improving over the $\lfloor\frac{2n}{3}\rfloor+1$ bound of Garijo et al. We then provide constructions of graphs reaching the $\frac{n}{2}$ bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees $G$ that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.Comment: 11 pages; 4 figure

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

Partitioning a graph into a dominating set, a total dominating set, and something else

A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph

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

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

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