The Domination and Independence of Some Cubic Bipartite Graphs

Abstract In this paper we discuss the relation between independent set and dominating set of finite simple graphs . In particular, we discuss them for some cubic bipartite graphs and find that the domination number is less than 1/3 of the number of vertices and independence number is half of the same. Mathematics Subject Classification: 05C6

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Offensive alliances in cubic graphs

An offensive alliance in a graph $\Gamma=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$'s closed neighborhood are in $S$. In the case of strong offensive alliance, strict majority is required. An alliance $S$ is called global if it affects every vertex in $V\backslash S$, that is, $S$ is a dominating set of $\Gamma$. The global offensive alliance number $\gamma_o(\Gamma)$ (respectively, global strong offensive alliance number $\gamma_{\hat{o}}(\Gamma)$) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in $\Gamma$. If $\Gamma$ has global independent offensive alliances, then the \emph{global independent offensive alliance number} $\gamma_i(\Gamma)$ is the minimum cardinality among all independent global offensive alliances of $\Gamma$. In this paper we study mathematical properties of the global (strong) alliance number of cubic graphs. For instance, we show that for all connected cubic graph of order $n$, $$\frac{2n}{5}\le \gamma_i(\Gamma)\le \frac{n}{2}\le \gamma_{\hat{o}}(\Gamma)\le \frac{3n}{4} \le \gamma_{\hat{o}}({\cal L}(\Gamma))=\gamma_{o}({\cal L}(\Gamma))\le n,$$ where ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. All the above bounds are tight

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

Some Results on incidence coloring, star arboricity and domination number

Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.Comment: 8 page