In the Cluster Vertex Deletion problem, we are given a graph G and an integer k, and the goal is to determine whether we can delete at most k vertices from G to make the remaining graph a cluster graph, i.e., a graph in which every connected component is a complete graph. In this paper, we show that Cluster Vertex Deletion can be solved in O^*(1.7549^k) time, improving the previous result of O^*(1.811^k) . To obtain this result, one crucial step is to show that Cluster Vertex Deletion on graphs of maximum degree at most 4 can be solved in O^*(1.7485^k) time. For a general graph, after a series of reductions, if the maximum degree of the reduced graph is at most 4, we introduce a new technique, called core branching processing, to solve the problem; if the reduced graph has a vertex of degree at least 5, we adopt the previous method of automated generation of search trees to obtain the improved running time.

Aggregating preferences of multiple entities is a problem that has been studied in various models of preference representation, including Conditional Preference Networks (CP-nets). Since optimal aggregation of CP-nets (for a specific natural choice of objective function) is known to require exponential time, efficient approximation algorithms have been proposed in the literature, yet with very limited results on the corresponding approximation ratio. In this paper, we show that a very simple and efficient method yields a 4/3-approximation for aggregating CP-nets from a proper superset of the set of all tree CP-nets-a well-studied class of CP-nets of relevance to many applications.

In this paper, we consider the edge searching problem and the fast searching problem with constraints on some vertices. The edge searching problem was introduced by Megiddo et al. (1988) [15], in which a group of searchers want to capture an invisible robber. The fast searching problem was introduced by Dyer et al. (2008) [8]. One constraint we consider for these two searching models is that a subset of vertices, called start vertices, are initially occupied by searchers. We want to find the minimum number of additional searchers to capture the robber. Another constraint is that there are some vertices, called halt vertices, which have to be occupied by searchers at the end of the searching process. For the edge searching on a tree containing only start vertices, we propose an O(n log n)-time algorithm for computing the edge search number, where n is the number of vertices. For the edge searching on a tree with start vertices and halt vertices, we give an O(n(2))-time algorithm to compute the edge search number. For the fast searching on a tree that contains only start vertices, we present a linear-time algorithm for computing the fast search number. For the fast searching on a tree with.. vertices that contains.. start vertices and h halt vertices, we propose an O((s + h)n)-time algorithm to compute the fast search number. For the edge searching (respectively, fast searching) with only halt vertices, we reduce the problem to the edge searching (respectively, fast searching) with start vertices.

Deduction is a recently introduced graph searching process in which searchers clear the vertex set of a graph with one move each, with each searcher's movement determined by which of its neighbors are protected by other searchers. In this paper, we show that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search. We give a structural characterization, new bounds and a spectrum result on the number of searchers required. We consider the complexity of computing this parameter, giving an NP-completeness result for arbitrary graphs, and exhibiting families of graphs for which the parameter can be computed in polynomial time. We also describe properties of the deduction process related to the timing of searcher movement and the success of terminal layouts.

The zero-visibility cops and robber game is a variant of Cops and Robbers subject to the constraint that the cops have no information at any time about the location of the robber. We first study a partition problem in which for a given graph and an integer k, we want to find a partition of the vertex set such that the size of the boundary of the smaller subset in the partition is at most k while the size of this subset is as large as possible under some conditions. Then we apply such partitions to prove lower bounds on the zero-visibility cop numbers of graph products. We also investigate the monotonic zero-visibility cop numbers of graph products.

This paper studies the design and analysis of approximation algorithms for aggregating preferences over combinatorial domains, represented using Conditional Preference Networks (CP-nets). Its focus is on aggregating preferences over so-called swaps, for which optimal solutions in general are already known to be of exponential size. We first analyze a trivial 2-approximation algorithm that simply outputs the best of the given input preferences, and establish a structural condition under which the approximation ratio of this algorithm is improved to 4/3. We then propose a polynomial-time approximation algorithm whose outputs are provably no worse than those of the trivial algorithm, but often substantially better. A family of problem instances is presented for which our improved algorithm produces optimal solutions, while, for any ε, the trivial algorithm cannot attain a (2- ε)-approximation. These results may lead to the first polynomial-time approximation algorithm that solves the CP-net aggregation problem for swaps with an approximation ratio substantially better than 2.

In the Cluster Vertex Deletion problem, we are given a graph G and an integer k, and the goal is to determine whether we can delete at most k vertices from G to make the remaining graph a cluster, i.e., a graph with each connected component being a complete graph. In this paper, we show that Cluster Vertex Deletion can be solved in $$O^*(1.7549^k)$$ time, improving the previous result of $$O^*(1.811^k)$$ . To obtain this result, one crucial step is to show Cluster Vertex Deletion on graphs of maximum degree at most 4 can be solved in $$O^*(1.7485^k)$$ time. After this, we know that the graph will always have a vertex of degree at least 5. Then by adopting the previous method of automated generation of searching trees, we can get the result on general graphs.

We consider zero-visibility cops and robber game that the cops lack of information on the location of the robber at all times, which is a variant of the classical cops and robbers game. First of all, we use the idea of splitting to study properties of cage graphs. Then we apply properties of cage graphs to investigate the lower bounds of cop number and the monotonic zero-visibility cop number of cage graphs. We also propose a searching algorithm to calculate the monotonic zero-visibility cop number of cage graphs.

Megiddo et al. (1988) introduced the edge searching problem, which is to find the minimum number of searchers to capture the robber in the edge searching model. Dyer et al. (2008) introduced the fast searching problem that is to find the minimum number of searchers to capture the robber in the fast searching model. In this paper, we consider these two graph searching problems under some constraints. One constraint is that a subset of vertices, called start vertices, are initially occupied by searchers before we place additional searchers on the graph. Another constraint is that some of the searchers must end their search at certain vertices called halt vertices. We focus on trees with n vertices. Let k be the number of times to move searchers from start vertices. For the edge searching problem, we give an O(kn)-time algorithm for computing the edge search number of a tree that contains only start vertices or only halt vertices. For a tree that contains both start vertices and halt vertices, we present an $$O(n^2)$$ -time algorithm to compute the edge search number. We show that all these problems are monotonic. For the fast searching problem, we propose a linear-time algorithm for computing the fast search number of a tree that contains only start vertices or only halt vertices. For a tree with n vertices that contains s start vertices and h halt vertices, we give an $$O((s+h)n)$$ -time algorithm to compute the fast search number.

The problem of finding the fast search number of a graph is NP-complete. It is challenging even when the graph has very small treewidth. However, it can be much easier to find an optimal fast search strategy for smaller subgraphs with special properties. This observation motivates us to establish relationships between optimal fast search strategies for a graph and its subgraphs although fast searching does not have the subgraph-closed property. In this paper, we introduce the notion of k -combinable graphs and study their properties. We propose a new method for computing the fast search number of k -combinable graphs. As an application of this method, we examine the fast searching for cactus graphs. We investigate the properties of optimal fast search strategies and give a linear time algorithm for computing the fast search number of cactus graphs.

In the one-visibility cops and robber game on a graph, the robber is visible to the cops only when the robber is in the closed neighbourhood of the vertices occupied by the cops. The one-visibility copnumber of a graph is the minimum number of cops required to capture the robber on the graph. In this paper, we investigate the one-visibility cops and robber game on trees. For trees, we introduce a key structure, called road, for characterising optimal cop-win strategies. We give an O (n logn) time algorithm to compute an optimal cop-win strategy for a tree with n vertices. We also establish relations between zero-visibility and one-visibility copnumbers on trees.(c) 2022 Elsevier B.V. All rights reserved.

This paper deals with constructing obstruction sets for two subclasses of 4-searchable graphs. We first characterize the 4-searchable biconnected outerplanar graphs by listing all graphs that cannot be their minors; we then give a constructive characterization of such graphs. We also characterize the 4-searchable biconnected generalized wheel graphs by listing all graphs that cannot be their minors.

Cops and Robbers is a classical pursuit and evasion game in graph theory, which was introduced by Nowakowski and Winkler and independently by Quilliot. In this paper, we study the zero-visibility cops and robber game, which is a variant of Cops and Robbers. In the zero-visibility cops and robber game, the robber is invisible to the cops throughout the game. We introduce a simple method for proving lower bounds on the zero-visibility cop number. This lower bound method is based on a connection between the zero-visibility cop number and the matching number. Using this technique, we investigate graph joins, lexicographic products of graphs, complete multipartite graphs and split graphs. For each of these classes of graphs, we prove lower bounds and upper bounds on the zero-visibility cop number. We also present a linear time approximation algorithm for computing the lexicographic product of a tree and a graph G. The approximation ratio of this algorithm is bounded by $$|V(G)| / (\nu (G) + |V(G) {\setminus } V(\mathcal {M}(G))| )$$ , where V(G) is the vertex set of G, $$\nu (G)$$ is the matching number of G, $$\mathcal {M}(G)$$ is a maximum matching of G and $$V(\mathcal {M}(G))$$ is the vertex set of $$\mathcal {M}(G)$$ .

The one-visibility cops and robber game is a variation of the classic cops and robber game, where one-visibility means that the information of the robber is known to all cops only when the distance between the robber and at least one cop is at most one. In this paper, we give a lower bound on the one-visibility copnumber of general trees. We present strategies to clear trees according to their structures. We propose a linear-time algorithm for computing the one-visibility copnumber of trees. (C) 2021 Elsevier B.V. All rights reserved.

This paper focuses on the task of aggregating preference orders over combinatorial domains, where both the individual and the aggregate preference orders are represented as Conditional Preference Networks (CP-nets). We propose intuitive objective functions for finding an optimal aggregate CP-net, as well as corresponding optimal efficient aggregation algorithms for inputs with certain structural properties.

AbstractWe correct an error in the original Lemma 3.4 in our paper “Achievable Multiplicity partitions in the IEVP of a graph”’ [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3 accordingly.

In this paper, we prove a lower bound for the one-visibility copnumber of trees. We give a linear-time algorithm for computing the one-visibility copnumber of trees. We also present relations between zero-visibility and one-visibility copnumbers on trees.

We correct an error in the original Lemma 3.4 in our paper "Achievable Multiplicity partitions in the IEVP of a graph"' [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3 accordingly.

This paper considers the one-cop-moves game played on a graph. In this game, a set of cops and a robber occupy the vertices of the graph and move alternately along the graph's edges with perfect information about each other's positions. The goal of the cops is to capture the robber. At cops' turns, exactly one cop is allowed to move from his location to an adjacent vertex; at robber's turns, she is allowed to move from her location to an adjacent vertex or to stay still. We want to find the minimum number of cops to capture the robber. This number is known as the cop number. In this paper, we investigate the cop number of several classes of graphs, including graphs with treewidth at most 2, Halin graphs, and Cartesian product graphs. We also give a characterization of k-winnable graphs in the one-cop-moves game. (C) 2020 Elsevier B.V. All rights reserved.

The LaTeX control sequence \deg is interpreted differently in the authors’ and the publisher’s LaTeX setting