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.

Minimum constraint removal problem for line segments is NP-hard

In the minimum constraint removal ([Formula: see text]), there is no feasible path to move from a starting point towards the goal, and the minimum constraints should be removed in order to find a collision-free path. It has been proved that [Formula: see text] problem is NP-hard when constraints have arbitrary shapes or even they are in shape of convex polygons. However, it has a simple linear solution when constraints are lines and the problem is open for other cases yet. In this paper, using a reduction from Subset Sum problem, in three steps, we show that the problem is NP-hard for both weighted and unweighted line segments.

On a spanning subgraph of the annihilating-ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. Let us denote the set of all annihilating ideals of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. With [Formula: see text], we associate an undirected graph, denoted by [Formula: see text], whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent in this graph if and only if [Formula: see text] and [Formula: see text]. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

On the double Roman bondage numbers of graphs

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text] for some vertex [Formula: see text], then [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text], and if [Formula: see text] then [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text]. The weight of a DRDF [Formula: see text] is the sum [Formula: see text]. The minimum weight of a DRDF on a graph [Formula: see text] is the double Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. The double Roman bondage number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all edge subsets [Formula: see text] such that [Formula: see text]. In this paper, we study the double Roman bondage number in graphs. We determine the double Roman bondage number in several families of graphs, and present several bounds for the double Roman bondage number. We also study the complexity issue of the double Roman bondage number and prove that the decision problem for the double Roman bondage number is NP-hard even when restricted to bipartite 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.

Birecognition of prime graphs, and minimal prime graphs

Given a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a module of [Formula: see text] if for each [Formula: see text], [Formula: see text] is adjacent to all the elements of [Formula: see text] or to none of them. For instance, [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text]) are the trivial modules of [Formula: see text]. A graph [Formula: see text] is prime if [Formula: see text] and all its modules are trivial. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given a graph [Formula: see text] such that [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-similar if for each [Formula: see text], [Formula: see text] and [Formula: see text] are both prime or not. The graph [Formula: see text] is said to be [Formula: see text]-birecognizable if every graph, [Formula: see text]-similar to [Formula: see text], is prime. We study the graphs [Formula: see text] that are not [Formula: see text]-birecognizable, where [Formula: see text] such that [Formula: see text] is prime, by using the following notion of a minimal prime graph. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given [Formula: see text], [Formula: see text] is [Formula: see text]-minimal if for each [Formula: see text] such that [Formula: see text], [Formula: see text] is not prime.

Strong chromatic index of generalized Jahangir graphs and generalized Helm graphs

The strong edge coloring of a graph G is a proper edge coloring that assigns a different color to any two edges which are at most two edges apart. The minimum number of color classes that contribute to such a proper coloring is said to be the strong chromatic index of G. This paper defines the strong chromatic index for the generalized Jahangir graphs and the generalized Helm graphs.

Injective edge-coloring of subcubic graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.