k-Efficient domination: Algorithmic perspective

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

ON HOP DOMINATION NUMBER OF SOME GENERALIZED GRAPH STRUCTURES

A subset \( H \subseteq V (G) \) of a graph \(G\) is a hop dominating set (HDS) if for every \({v\in (V\setminus H)}\) there is at least one vertex \(u\in H\) such that \(d(u,v)=2\). The minimum cardinality of a hop dominating set of \(G\) is called the hop domination number of \(G\) and is denoted by \(\gamma_{h}(G)\). In this paper, we compute the hop domination number for triangular and quadrilateral snakes. Also, we analyse the hop domination number of graph families such as generalized thorn path, generalized ciliates graphs, glued path graphs and generalized theta graphs.

2-Point set domination in separable graphs

In a graph [Formula: see text], a set [Formula: see text] is a [Formula: see text]-point set dominating set (in short 2-psd set) of [Formula: see text] if for every subset [Formula: see text] there exists a nonempty subset [Formula: see text] containing at most two vertices such that the induced subgraph [Formula: see text] is connected in [Formula: see text]. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. The main focus of this paper is to find the value of [Formula: see text] for a separable graph and thereafter computing [Formula: see text] for some well-known classes of separable graphs. Further we classify the set of all 2-psd sets of a separable graph into six disjoint classes and study the existence of minimum 2-psd sets in each class.

Domination and Independent Domination in Hexagonal Systems

A vertex subset D of G is a dominating set if every vertex in V(G)∖D is adjacent to a vertex in D. A dominating set D is independent if G[D], the subgraph of G induced by D, contains no edge. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G, and the independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. A classical work related to the relationship between γ(G) and i(G) of a graph G was established in 1978 by Allan and Laskar. They proved that every K1,3-free graph G satisfies γ(G)=i(H). Hexagonal systems (2 connected planar graphs whose interior faces are all hexagons) have been extensively studied as they are used to present bezenoid hydrocarbon structures which play an important role in organic chemistry. The domination numbers of hexagonal systems have been studied continuously since 2018 when Hutchinson et al. posted conjectures, generated from a computer program called Conjecturing, related to the domination numbers of hexagonal systems. Very recently in 2021, Bermudo et al. answered all of these conjectures. In this paper, we extend these studies by considering the relationship between the domination number and the independent domination number of hexagonal systems. Although every hexagonal system H with at least two hexagons contains K1,3 as an induced subgraph, we find many classes of hexagonal systems whose domination number is equal to an independent domination number. However, we establish the existence of a hexagonal system H such that γ(H)<i(H) with the prescribed number of hexagons.

Upper paired domination versus upper domination

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

Edge-vertex domination in trees

Let [Formula: see text] be a finite simple graph. A vertex [Formula: see text] is edge-vertex dominated by an edge [Formula: see text] if [Formula: see text] is incident with [Formula: see text] or [Formula: see text] is incident with a vertex adjacent to [Formula: see text]. An edge-vertex dominating set of [Formula: see text] is a subset [Formula: see text] such that every vertex of [Formula: see text] is edge-vertex dominated by an edge of [Formula: see text]. The edge-vertex domination number [Formula: see text] is the minimum cardinality of an edge-vertex dominating set of [Formula: see text]. In this paper, we prove that [Formula: see text] for every tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, and we characterize the trees attaining each of the bounds.

On some variations of dominating identification in graphs

In this study, we introduce the locating-dominating value and the location-domination polynomial of graphs and location-domination polynomials of some families of graphs were identified. Locatingdominating set of graph G is defined as the dominating set which locates all the vertices of G. And, location-domination number G is the minimum cardinality of a locating-dominating set in G.