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Given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, we investigate the 2-Eigenvalue Vertex Deletion (2-EVD) problem. The objective is to remove at most $k$ vertices such that the adjacency matrix of the resulting graph has at most two eigenvalues. It is established that the adjacency matrix of a graph has at most two eigenvalues if and only if the graph is a collection of equal-sized cliques. Thus, the 2-Eigenvalue Vertex Deletion amounts to removing a set of at most $k$ vertices to transform the graph into a collection of equal-sized cliques. The 2-Eigenvalue Edge Editing (2-EEE), 2-Eigenvalue Edge Deletion (2-EED) and 2-Eigenvalue Edge Addition (2-EEA) problems are defined analogously. We present a kernel of size $\mathcal{O}(k^{3})$ for $2$-EVD, along with an FPT algorithm with a running time of $\mathcal{O}^{*}(2^{k})$. For the problem $2$-EEE, we provide a kernel of size $\mathcal{O}(k^{2})$. Additionally, we present linear kernels of size $5k$ and $6k$ for $2$-EEA and $2$-EED respectively. For the $2$-EED, we also construct an algorithm with running time $\mathcal{O}^{*}(1.47^{k})$ . These results address open questions posed by Misra et al. (ISAAC 2023) regarding the complexity of these problems when parameterized by the solution size.

A knot $K$ in a directed graph $D$ is a strongly connected component of size at least two such that there is no arc $(u,v)$ with $u \in V(K)$ and $v\notin V(K)$. Given a directed graph $D=(V,E)$, we study Knot-Free Vertex Deletion (KFVD), where the goal is to remove the minimum number of vertices such that the resulting graph contains no knots. This problem naturally emerges from its application in deadlock resolution since knots are deadlocks in the OR-model of distributed computation. The fastest known exact algorithm in literature for KFVD runs in time $\mathcal{O}^\star(1.576^n)$. In this paper, we present an improved exact algorithm running in time $\mathcal{O}^\star(1.4549^n)$, where $n$ is the number of vertices in $D$. We also prove that the number of inclusion wise minimal knot-free vertex deletion sets is $\mathcal{O}^\star(1.4549^n)$ and construct a family of graphs with $\Omega(1.4422^n)$ minimal knot-free vertex deletion sets

A set $D$ of vertices of a graph is a \emph{defensive alliance} if, for each element of $D$, the majority of its neighbours are in $D$. We consider the notion of local minimality in this paper. We are interested in finding a locally minimal defensive alliance of maximum size. In Locally Minimal Defensive Alliance problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a locally minimal defensive alliance of size at least $k$. This problem is known to be NP-hard, but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance restricted to the graphs of minimum degree at least 2 is fixed-parameter tractable (FPT) when parameterized by the combined parameters solution size $k$, and maximum degree $\Delta$ of the input graph, (2) Locally Minimal Defensive Alliance on the graphs of minimum degree at least 2, admits a kernel with at most $k^{k^{\mathcal{O}(k)}}$ vertices. In particular, the problem parameterized by $k$ restricted to $C_3$-free and $C_4$-free graphs of minimum degree at least 2, admits a kernel with at most $k^{\mathcal{O}(k)}$ vertices. Moreover, we prove that the problem on planar graphs of minimum degree at least 2, admits an FPT algorithm with running time $\mathcal{O}^{*}(k^{2^{\mathcal{O}(\sqrt{k})}})$. Finally, we prove that (4) Locally Minimal Defensive Alliance Extension is NP-complete., Comment: arXiv admin note: text overlap with arXiv:2105.10742

A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the classical and parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on natural parameter as well as parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; we thereby resolve an open question stated by Bernhard Bliem and Stefan Woltran (2018) concerning the complexity of Offensive Alliance parameterized by treewidth, (2) unless ETH fails, Offensive Alliance problem cannot be solved in time $\mathcal{O}^{*}(2^{o(k \log k)})$ where $k$ is the solution size, (3) Offensive Alliance problem does not admit a polynomial kernel parameterized by solution size and vertex cover of the input graph. On the positive side we prove that (4) Offensive Alliance can be solved in time $\mathcal{O}^{*}(\tt{vc(G)}^{\mathcal{O}(\tt{vc(G)})})$ where $\tt{vc(G)}$ is the vertex cover number of the input graph. In terms of classical complexity, we prove that (5) Offensive Alliance problem cannot be solved in time $2^{o(n)}$ even when restricted to bipartite graphs, unless ETH fails, (6) Offensive Alliance problem cannot be solved in time $2^{o(\sqrt{n})}$ even when restricted to apex graphs, unless ETH fails. We also prove that (7) Offensive Alliance problem is NP-complete even when restricted to bipartite, chordal, split and circle graphs., Comment: arXiv admin note: substantial text overlap with arXiv:2110.15757

In this paper we study a maximization version of the classical Edge Dominating Set (EDS) problem, namely, the Upper EDS problem, in the realm of Parameterized Complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal edge dominating set of size at least $k$. We obtain the following results for Upper EDS. We prove that Upper EDS admits a kernel with at most $4k^2-2$ vertices. We also design a fixed-parameter tractable (FPT) algorithm for Upper EDS running in time $2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$.

In this paper we study a maximization version of the classical Feedback Vertex Set (FVS) problem, namely, the Max Min FVS problem, in the realm of parameterized complexity. In this problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a minimal feedback vertex set of size at least $k$. We obtain following results for Max Min FVS. 1) We first design a fixed parameter tractable (FPT) algorithm for Max Min FVS running in time $10^kn^{\mathcal{O}(1)}$. 2) Next, we consider the problem parameterized by the vertex cover number of the input graph (denoted by $\mathsf{vc}(G)$), and design an algorithm with running time $2^{\mathcal{O}(\mathsf{vc}(G)\log \mathsf{vc}(G))}n^{\mathcal{O}(1)}$. We complement this result by showing that the problem parameterized by $\mathsf{vc}(G)$ does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. 3) Finally, we give an FPT-approximation scheme (fpt-AS) parameterized by $\mathsf{vc}(G)$. That is, we design an algorithm that for every $\epsilon >0$, runs in time $2^{\mathcal{O}\left(\frac{\mathsf{vc}(G)}{\epsilon}\right)} n^{\mathcal{O}(1)}$ and returns a minimal feedback vertex set of size at least $(1-\epsilon){\sf opt}$.

A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider the notion of global minimality in this paper. We are interested in globally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the Globally Minimal Defensive Alliance problem is FPT parameterized by the neighbourhood diversity of the input graph. The result for neighborhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treewidth and treedepth. We also proved that, given a vertex $r \in V(G)$, deciding if $G$ has a globally minimal defensive alliance of any size containing vertex $r$ is NP-complete., Comment: arXiv admin note: text overlap with arXiv:2105.10742

Given a graph $G = (V,E)$, a threshold function $t~ :~ V \rightarrow \mathbb{N}$ and an integer $k$, we study the Harmless Set problem, where the goal is to find a subset of vertices $S \subseteq V$ of size at least $k$ such that every vertex $v\in V$ has less than $t(v)$ neighbors in $S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity. Our focus lies on parameters that measure the structural properties of the input instance. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number. We also show that the Harmless Set problem with majority thresholds is W[1]-hard when parameterized by the treewidth of the input graph. We prove that the Harmless Set problem can be solved in polynomial time on graph with bounded cliquewidth. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to neighbourhood diversity, twin cover and vertex integrity of the input graph. We show that the problem parameterized by the solution size is fixed parameter tractable on planar graphs. We thereby resolve two open questions stated in C. Bazgan and M. Chopin (2014) concerning the complexity of {\sc Harmless Set} parameterized by the treewidth of the input graph and on planar graphs with respect to the solution size.

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours are in $S$. We study the parameterized complexity of the Defensive Alliance problem, where the aim is to find a minimum size defensive alliance. Our main results are the following: (1) The Defensive Alliance problem has been studied extensively during the last twenty years, but the question whether it is FPT when parameterized by feedback vertex set has still remained open. We prove that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; (2) the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP $\subseteq$ NP/poly, (3) it does not admit $2^{o(n)}$ algorithm under ETH, and (4) the Defensive Alliance problem on circle graphs is NP-complete., Comment: arXiv admin note: text overlap with arXiv:2110.15757

The Offensive Alliance problem has been studied extensively during the last twenty years. A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph.

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. We also look at connected version of defensive alliance. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance is NP-complete, even when restricted to planar graphs, (2) a randomized FPT algorithm for Exact Connected Locally Minimal Defensive Alliance parameterized by solution size, (3) Locally Minimal Defensive Alliance is fixed-parameter tractable (FPT) when parametrized by neighbourhood diversity, (4) Locally Minimal Defensive Alliance parameterized by treewidth is W[1]-hard and thus not FPT (unless FPT=W[1]), (5) Locally Minimal Defensive Alliance can be solved in polynomial time for graphs of bounded treewidth.

Given a graph $G=(V,E)$, a set $\mathcal{F}$ of forbidden subgraphs, we study $\mathcal{F}$-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any $F\in \mathcal{F}$ as a subgraph. For the parameter treewidth, the question of whether the problem is FPT has remained open. Here we give a negative answer by showing that the problem is W[1]-hard when parameterized by the treewidth, which rules out FPT algorithms under common assumption. Thus we give a solution to the conjecture posted by Jessica Enright and Kitty Meeks in [Algorithmica 80 (2018) 1857-1889]. We also prove that the $\mathcal{F}$-Free Edge Deletion problem is W[2]-hard when parameterized by the solution size $k$, feedback vertex set number or pathwidth of the input graph. A special case of particular interest is the situation in which $\mathcal{F}$ is the set $\mathcal{T}_{h+1}$ of all trees on $h+1$ vertices, so that we delete edges in order to obtain a graph in which every component contains at most $h$ vertices. This is desirable from the point of view of restricting the spread of disease in transmission network. We prove that the $\mathcal{T}_{h+1}$-Free Edge Deletion problem is fixed-parameter tractable (FPT) when parameterized by the vertex cover number. We also prove that it admits a kernel with $O(hk)$ vertices and $O(h^2k)$ edges, when parameterized by combined parameters $h$ and the solution size $k$.

The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part. This problem was introduced by Gerber and Kobler [European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors, but its parameterized complexity remains open until now. It is known that the Satisfactory Partition problem, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is FPT when parameterized by the neighbourhood diversity of the input graph, (2) it can be solved in $O(n^{8 {\tt cw}})$ where ${\tt cw}$ is the clique-width,(3) a generalized version of the problem is W[1]-hard when parameterized by the treewidth., Comment: 1-15 pages, 2 figures

Two vectors $x,y$ in $\mathbb{Z}_g^n$ are $ qualitatively$ $ independent$ if for all pairs $(a,b)\in \mathbb{Z}_g\times \mathbb{Z}_g$, there exists $i\in \{1,2,\ldots,n\}$ such that $(x_i,y_i)=(a,b)$. A covering array on a graph $G$, denoted by $CA(n,G,g)$, is a $|V(G)|\times n$ array on $\mathbb{Z}_g$ with the property that any two rows which correspond to adjacent vertices in $G$ are qualitatively independent. The number of columns in such array is called its $size$. Given a graph $G$, a covering array on $G$ with minimum size is called $optimal$. Our primary concern in this paper is with constructions that make optimal covering arrays on large graphs those are obtained from product of smaller graphs. We consider four most extensively studied graph products in literature and give upper and lower bounds on the the size of covering arrays on graph products. We find families of graphs for which the size of covering array on the Cartesian product achieves the lower bound. Finally, we present a polynomial time approximation algorithm with approximation ratio $\log(\frac{V}{2^{k-1}})$ for constructing covering array on graph $G=(V,E)$ with $k>1$ prime factors with respect to the Cartesian product.

A covering array $t$-$CA(n,k,g)$, of size $n$, strength $t$, degree $k$, and order $g$, is a $k\times n$ array on $g$ symbols such that every $t\times n$ sub-array contains every $t\times 1$ column on $g$ symbols at least once. Covering arrays have been studied for their applications to software testing, hardware testing, drug screening, and in areas where interactions of multiple parameters are to be tested. In this paper, we present an algebraic construction that improves many of the best known upper bounds on $n$ for covering arrays 4-$CA(n,k,g)$ with $g=3$. The $coverage$ $measure$ $\mu_t(A)$ of a testing array $A$ is defined by the ratio between the number of distinct $t$-tuples contained in the column vectors of $A$ and the total number of $t$-tuples. A covering array is a testing array with full coverage. The $covering$ $arrays$ $with$ $budget$ $constraints$ $problem$ is the problem of constructing a testing array of size at most $n$ having largest possible coverage measure, given values of $k,g$ and $n$. This paper presents several strength four testing arrays with high coverage. The construction here is a generalisation of the construction methods used by Chateauneuf, Colbourn and Kreher, and Meagher and Stevens.

Covering arrays are combinatorial objects that have been successfully applied in the design of test suites for testing systems such as software, circuits and networks, where failures can be caused by the interaction between their parameters. In this paper, we perform a new generalization of covering arrays called covering arrays on 3-uniform hypergraphs. Let $n, k$ be positive integers with $k\geq 3$. Three vectors $x\in \mathbb Z_{g_1}^n$, $y\in \mathbb Z_{g_2}^n$, $z\in \mathbb Z_{g_3}^n$ are {\it 3-qualitatively independent} if for any triplet $(a, b, c) \in \mathbb Z_{g_1}\,\times\, \mathbb Z_{g_2}\,\times\,\mathbb Z_{g_3}$, there exists an index $ j\in \lbrace 1, 2,...,n \rbrace $ such that $( x(j), y(j), z(j)) = (a, b, c)$. Let $H$ be a 3-uniform hypergraph with $k$ vertices $v_1,v_2,\ldots,v_k$ with respective vertex weights $g_1,g_2,\ldots,g_k$. A mixed covering array on $H$, denoted by $3-CA(n,H, \prod_{i=1}^{k}g_{i})$, is a $k\times n$ array such that row $i$ corresponds to vertex $v_i$, entries in row $i$ are from $Z_{g_i}$; and if $\{v_x,v_y,v_z\}$ is a hyperedge in $H$, then the rows $x,y,z$ are 3-qualitatively independent. The parameter $n$ is called the size of the array. Given a weighted 3-uniform hypergraph $H$, a mixed covering array on $H$ with minimum size is called optimal. We outline necessary background in the theory of hypergraphs that is relevant to the study of covering arrays on hypergraphs. In this article, we introduce five basic hypergraph operations to construct optimal mixed covering arrays on hypergraphs. Using these operations, we provide constructions for optimal mixed covering arrays on $\alpha$-acyclic 3-uniform hypergraphs, conformal 3-uniform hypertrees having a binary tree as host tree, and on some specific 3-uniform cycle hypergraphs.