General dd-position sets

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural characterization of general $d$-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is infinite whenever $G$ is an infinite graph and $d$ is a finite integer.Comment: 16 page

Generalization of edge general position problem

The edge geodesic cover problem of a graph $G$ is to find a smallest number of geodesics that cover the edge set of $G$. The edge $k$-general position problem is introduced as the problem to find a largest set $S$ of edges of $G$ such that no $k-1$ edges of $S$ lie on a common geodesic. We study this dual min-max problems and connect them to an edge geodesic partition problem. Using these connections, exact values of the edge $k$-general position number is determined for different values of $k$ and for different networks including torus networks, hypercubes, and Benes networks.Comment: This research is supported by Kuwait University, Kuwai

Geodesic packing in graphs

Given a graph $G$, a geodesic packing in $G$ is a set of vertex-disjoint maximal geodesics, and the geodesic packing number of $G$, {\gpack}(G), is the maximum cardinality of a geodesic packing in $G$. It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, ${\gt}(G)$, which is the minimum cardinality of a set of vertices that hit all maximal geodesics in $G$. While \gt(G)\ge \gpack(G) in every graph $G$, the quotient ${\rm gt}(G)/{\rm gpack}(G)$ is investigated. By using the rook's graph, it is proved that there does not exist a constant $C < 3$ such that $\frac{{\rm gt}(G)}{{\rm gpack}(G)}\le C$ would hold for all graphs $G$. If $T$ is a tree, then it is proved that ${\rm gpack}(T) = {\rm gt}(T)$, and a linear algorithm for determining ${\rm gpack}(T)$ is derived. The geodesic packing number is also determined for the strong product of paths

Variety of mutual-visibility problems in graphs

If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.Comment: 23 pages, 4 figures, original pape

Grundy dominating sequences and zero forcing sets

In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. The length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities

The domination game played on diameter 2 graphs

Let gamma(g)(G) be the game domination number of a graph G. It is proved that if diam(G) = 2, then gamma(g)(G) <= inverted right perpendicularn(G)/2inverted left perpendicular - left perpendicularn(G)/11right perpendicular. The bound is attained: if diam(G) = 2 and n(G) <= 10, then gamma(g)(G) = inverted right perpendicularn(G)/2inverted left perpendicular if and only if G is one of seven sporadic graphs with n(G) = 6 or the Petersen graph, and there are exactly ten graphs of diameter 2 and order 11 that attain the bound