Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real number $α$ with $0<α\leq1$, a set $D\subseteq V$ is called an $α$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $α\times d_v$ neighbors in $D$, where $d_v$ is the degree of $v$ in $G$. The cardinality of a minimum $k$-dominating set and a minimum $α$-dominating set in $G$ is said to be the $k$-domination number and the $α$-domination number of $G$, respectively. In this paper, we present some approximability and inapproximability results on the problem of finding $k$-domination number and $α$-domination number of some classes of graphs. Moreover, we introduce a generalization of $α$-dominating set which we call an $f$-dominating set. Given a function $f:\mathbb{N}\rightarrow \mathbb{R}$, where $\mathbb{N}=\{1, 2, 3, \ldots\}$, a set $D\subseteq V$ is said to be an $f$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $f(d_v)$ neighbors in $D$. We prove NP-hardness of the problem of finding a minimum $f$-dominating set in $G$, for a large family of functions $f$.
The game domination number is a graph invariant that arises from a game, which is related to graph domination in a similar way as the game chromatic number is related to graph coloring. In this paper we show that verifying whether the game domination number of a graph is bounded by a given integer is PSPACE-complete. This contrasts the situation of the game coloring problem whose complexity is still unknown.
Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. A dominating set $D$ is called a total dominating set if every vertex in $D$ is adjacent to a vertex in $D$. The domination (resp. total domination) number of $G$ is the smallest cardinality of a dominating (resp. total dominating) set of $G$. The bondage (resp. total bondage) number of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination (resp. total domination) number of $G$. The reinforcement number …
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected (total) dominating set $S \subseteq V$ is a secure connected (total) dominating set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such that $uv \in E$ and $(S \setminus \lbrace v \rbrace) \cup \lbrace u \rbrace $ is a connected (total) dominating set of $G$. The minimum cardinality of a secure connected (total) dominating set of $G$ denoted by $ γ_{sc} (G) (γ_{st}(G))$, is called the secure connected (total) domination number of $G$. In this paper, we show that the decision problems corresponding to s…
We study the parameterized complexity of domination-type problems. (sigma,rho)-domination is a general and unifying framework introduced by Telle: a set D of vertices of a graph G is (sigma,rho)-dominating if for any v in D, |N(v)\cap D| in sigma and for any $v\notin D, |N(v)\cap D| in rho. We mainly show that for any sigma and rho the problem of (sigma,rho)-domination is W[2] when parameterized by the size of the dominating set. This general statement is optimal in the sense that several particular instances of (sigma,rho)-domination are W[2]-complete (e.g. Dominating Set). We also prove that…
In this paper, we study the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by at least one using $k$ edge contractions, for some fixed integer $k \geq 0$? We present positive and negative results regarding the computational complexity of this problem.
We generalize the family of $(σ, ρ)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as $k$-trapezoid graphs, Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, $k$-polygon graphs,…
A set $S \subseteq V$ is a dominating set in G if for every u \in V \ S, there exists $v \in S$ such that $(u,v) \in E$, i.e., $N[S] = V$. A dominating set $S$ is an Isolate Dominating Set} (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set $S \subseteq V$ is an independent set if G[S] has no edge. A set S \subsete…
A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set $S\subseteq V$ is an isolate secure dominating set (ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ is an IDS of $G$. The minimum cardinality of an ISDS of $G$ is called as an isolate secure domination number, and is denoted by $γ_{0s}(G)$. Given a graph $ …
Identifying and mitigating the spread of fake information is a challenging issue that has become dominant with the rise of social media. We consider a generalization of the Domination problem that can be used to detect a set of individuals who, once immunized, can prevent the spreading of fake narratives. The considered problem, named {\em Distance Vector Domination} generalizes both distance and multiple domination, at individual (i.e., vertex) level. We study the parameterized complexity of the problem according to several standard and structural parameters. We prove the W[1]-hardness of the…
For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision …
In this paper, we study the parameterized complexity of a generalized domination problem called the [$σ, ρ$] Dominating Set problem. This problem generalizes a large number of problems including the Minimum Dominating Set problem and its many variants. The parameterized complexity of the [$σ, ρ$] Dominating Set problem parameterized by treewidth is well studied. Here the properties of the sets $σ$ and $ρ$ that make the problem tractable are identified [1]. We consider a larger parameter and investigate the existence of polynomial sized kernels. When $σ$ and $ρ$ are finite, we identify the exac…
In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and minimisation problems, as well as the related parameterised problems, on general graphs and on graphs of bounded degree, and we also study planar graphs.
We study the complexity of the two dual covering and packing distance-based problems Broadcast Domination and Multipacking in digraphs. A dominating broadcast of a digraph $D$ is a function $f:V(D)\to\mathbb{N}$ such that for each vertex $v$ of $D$, there exists a vertex $t$ with $f(t)>0$ having a directed path to $v$ of length at most $f(t)$. The cost of $f$ is the sum of $f(v)$ over all vertices $v$. A multipacking is a set $S$ of vertices of $D$ such that for each vertex $v$ of $D$ and for every integer $d$, there are at most $d$ vertices from $S$ within directed distance at most $d$ from $…
Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed for each interval - one from the left and one from the right side. Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complex…
In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time $\frac{11}{2}$-factor approximation algorithm \cite{bhore} for the MLDS problem is erroneous and propose a simple $O(n + m)$ time 7.31-factor approximation algorithm, where $n$ and $m$ are the number of vertices and edges in the input unit disk graph, respectively. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.
Given a graph $G=(V,E)$, a subset $B\subseteq V$ of vertices is a weak odd dominated (WOD) set if there exists $D \subseteq V {\setminus} B$ such that every vertex in $B$ has an odd number of neighbours in $D$. $κ(G)$ denotes the size of the largest WOD set, and $κ'(G)$ the size of the smallest non-WOD set. The maximum of $κ(G)$ and $|V|-κ'(G)$, denoted $κ_Q(G)$, plays a crucial role in quantum cryptography. In particular deciding, given a graph $G$ and $k>0$, whether $κ_Q(G)\le k$ is of practical interest in the design of graph-based quantum secret sharing schemes. The decision problems assoc…
We show that the problem k-Dominating Set and its several variants including k-Connected Dominating Set, k-Independent Dominating Set, and k-Dominating Clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-Distance k-Dominating Set for d >= 2 is W[2]-complete in multiple-interval graphs and their complements. We…
The concept of generalized domination unifies well-known variants of domination-like and independence problems, such as Dominating Set, Independent Set, Perfect Code, etc. A generalized domination (also called $[σ,ρ]$-Dominating Set}) problem consists in finding a subset of vertices in a graph such that every vertex is satisfied with respect to two given sets of constraints $σ$ and $ρ$. Very few problems are known not to be FPT when parameterized by tree-width, as usually this restriction allows one to write efficient algorithms to solve the considered problems. The main result of this article…
A semitotal dominating set of a graph $G$ with no isolated vertex is a dominating set $D$ of $G$ such that every vertex in $D$ is within distance two of another vertex in $D$. The minimum size $γ_{t2}(G)$ of a semitotal dominating set of $G$ is squeezed between the domination number $γ(G)$ and the total domination number $γ_{t}(G)$. \textsc{Semitotal Dominating Set} is the problem of finding, given a graph $G$, a semitotal dominating set of $G$ of size $γ_{t2}(G)$. In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph c…
Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in $S$. A set $\tilde{D} \subseteq V$ of a graph $G=(V,E) $ is called an outer-connected dominating set for $G$ if (1) $\tilde{D}$ is a dominating set for $G$, and (2) $G [V \setminus \tilde{D}]$, the induced subgraph of $G$ by $V \setminus \tilde{D}$, is connected. The minimum size among all outer-connected dominating sets of $G$ is called the outer-connected domination number of $G$ and is denoted by $\tildeγ_c(G)$. We define the outer-connected bondage number of a…
Given a graph $G$ with a terminal set $R \subseteq V(G)$, the Steiner tree problem (STREE) asks for a set $S\subseteq V(G) \setminus R$ such that the graph induced on $S\cup R$ is connected. A split graph is a graph which can be partitioned into a clique and an independent set. It is known that STREE is NP-complete on split graphs \cite{white1985steiner}. To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique ($K$), whereas STREE is NP-co…
The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted $\ell_1$ norm over the solution set of a linear complementarity problem (LCP). The minimum $\ell_1$ norm of solutions of this LCP is a lower bound on the independent domination number of the graph. Unlike the case of the maximum $\ell_1$ norm, this lower bound is in ge…
A set $D\subseteq V$ is called a $k$-tuple dominating set of a graph $G=(V,E)$ if $\left| N_G[v] \cap D \right| \geq k$ for all $v \in V$, where $N_G[v]$ denotes the closed neighborhood of $v$. A set $D \subseteq V$ is called a liar's dominating set of a graph $G=(V,E)$ if (i) $\left| N_G[v] \cap D \right| \geq 2$ for all $v\in V$ and (ii) for every pair of distinct vertices $u, v\in V$, $\left| (N_G[u] \cup N_G[v]) \cap D \right| \geq 3$. Given a graph $G$, the decision versions of $k$-Tuple Domination Problem and the Liar's Domination Problem are to check whether there exists a $k$-tuple dom…
A set $D \subseteq V$ of a graph $G = (V,E)$ is called an outer-connected dominating set of $G$ if every vertex $v$ not in $D$ is adjacent to at least one vertex in $D$, and the induced subgraph of $G$ on $V \setminus D$ is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality for the input graph $G$. Given a positive integer $k$ and a graph $G = (V, E)$, the Outer-connected Domination Decision problem is to decide whether $G$ has an outer-connected dominating set of cardinality at most $k$. The Outer-connected Domination …
Roman domination is one of few examples where the related extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting, as it allows to establish polynomial-delay enumeration algorithms for finding minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets of a graph or if all hitting sets of a hypergraph can be enumerated with polynomial delay. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain and study two …
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \} $ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $ γ_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $ G$ and a positive integer $ k,$ the Secure Connected Domination (SCDM) problem is to check whether $ G $ has a secure conn…
A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: - Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, param…
We consider the problem of reducing the (semi)total domination number of graph by one by contracting edges. It is known that this can always be done with at most three edge contractions and that deciding whether one edge contraction suffices is an $\mathsf{NP}$-hard problem. We show that for every fixed $k \in \{2,3\}$, deciding whether exactly $k$ edge contractions are necessary is $\mathsf{NP}$-hard and further provide for $k=2$ complete complexity dichotomies on monogenic graph classes.
In this paper we are going to solve an open problem about the game tetris. We are going to give the first results in the complexity of a variant of offline tetris introduced by Erik Demaine, Susan Hohenberger and David Liben Nowell in their paper "Tetris is hard, even to approximate". In this variant, that follows a model of movements introduced by John Brzustowsky, we can move and rotate a piece the number of times we want in the first row. But then, when we left the piece fall, we cannot move it or rotate it anymore. We are going to demonstrate that the problem of maximizing the number of cl…
