Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two... more Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. We are interested in giving upper bounds on the vertex bisection width of random d-regular graphs for constant values of d. Our approach is based on analyzing a greedy algorithm by using the Differential Equations Method. In this way, we obtain the first known upper bounds for the vertex bisection width in random regular graphs. The results are compared with experimental ones and with lower bounds obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers
List k-Coloring (LI k-COL) is the decision problem asking if a given graph admits a proper colori... more List k-Coloring (LI k-COL) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1, 2,. . ., k}. The problem is known to be NP-hard even for k = 3 within the class of 3-regular planar bipartite graphs and for k = 4 within the class of chordal bipartite graphs. In 2015 Huang, Johnson and Paulusma asked for the complexity of LI 3-COL in the class of chordal bipartite graphs. In this paper we give a partial answer to this question by showing that LI k-COL is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the LI k-COL in the calss of convex bipartite graphs. Finally we show how our algorithm can be modified to solve the more general LI H-COL problem on convex bipartite graphs.
The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a serie... more The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops on theory and applications of discrete algorithms, graphs and combinatorial structures in the wide sense, organized by either
We consider the following problem: can a certain graph parameter of some given graph be reduced b... more We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pfree graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomialtime solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.
Edge searching is a graph problem that corresponds to cleaning a contaminated graph using the min... more Edge searching is a graph problem that corresponds to cleaning a contaminated graph using the minimum number of searchers. We define fast searching as a variant of this widely studied problem. Fast searching corresponds to an internal monotone search in which every edge is traversed exactly once and searchers are not allowed to jump. We present a linear time algorithm to compute the fast search number of trees. We investigate the fast search number of bipartite graphs. We also propose a general cost function for evaluating search strategies that utilizes both edge searching and fast searching.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:
We introduce the multicolored graph realization problem (MGR). The input to the problem is a colo... more We introduce the multicolored graph realization problem (MGR). The input to the problem is a colored graph (G,φ), i.e., a graph together with a coloring φ on its vertices. We can associate to each colored graph (G,φ) a cluster graph (Gφ) in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices S is multicolored when S has exactly one vertex from each color class. The problem is to decide whether there is a multicolored set S such that, after identifying each vertex in S with its color class, G[S] coincides with Gφ. The MGR problem is related to the well known class of generalized network problems, most of which are NP-hard. For example the generalized Minimun Spanning Tree problem. The MGR problem is a generalization of the multicolored clique problem, which is known to be W [1]-hard when parameterized by the number of colors. Thus MGR remains W [1]-hard, when parameterized by the size of the cluster graph. Thi...
We study the following problem: for given integers d, k and graph G, can we reduce some fixed gra... more We study the following problem: for given integers d, k and graph G, can we reduce some fixed graph parameter π of G by at least d via at most k graph operations from some fixed set S? As parameters we take the chromatic number χ, clique number ω and independence number α, and as operations we choose the edge contraction ec and vertex deletion vd. We determine the complexity of this problem for S = {ec} and S = {vd} and π ∈ {χ, ω, α} for a number of subclasses of perfect graphs. We use these results to determine the complexity of the problem for S = {ec} and S = {vd} and π ∈ {χ, ω, α} restricted to H-free graphs.
In the edge searching problem, searchers move from vertex to vertex in a graph to capture an invi... more In the edge searching problem, searchers move from vertex to vertex in a graph to capture an invisible, fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which, at every move, a new edge of the graph G must be guaranteed to be free of the intruder. That is, once all searchers are placed the graph G is cleared in exactly |E(G)| moves. Such a restriction obviously necessitates a larger number of searchers. We examine this model, and characterize graphs for which 2 or 3 searchers are sufficient. We prove that the corresponding decision problem is NP-complete.
ABSTRACT We consider the following problem: can a certain graph parameter of some given graph be ... more ABSTRACT We consider the following problem: can a certain graph parameter of some given graph be reduced by at most $d$ for some integer~$d$ via at most $k$ edge contractions for some given integer $k$? We consider three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when $d$ is part of the input, this problem is polynomial-time solvable on $P_4$-free graphs and \NP-complete for split graphs. Moreover, as split graphs form a subclass of $P_5$-free graphs, both results together give a complete complexity classification for $P_\ell$-free graphs. On the positive side, for split graphs, we also show that the problem is polynomial-time solvable if $d$ is fixed, that is, not part of the input.
Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two... more Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. We are interested in giving upper bounds on the vertex bisection width of random d-regular graphs for constant values of d. Our approach is based on analyzing a greedy algorithm by using the Differential Equations Method. In this way, we obtain the first known upper bounds for the vertex bisection width in random regular graphs. The results are compared with experimental ones and with lower bounds obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers
List k-Coloring (LI k-COL) is the decision problem asking if a given graph admits a proper colori... more List k-Coloring (LI k-COL) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1, 2,. . ., k}. The problem is known to be NP-hard even for k = 3 within the class of 3-regular planar bipartite graphs and for k = 4 within the class of chordal bipartite graphs. In 2015 Huang, Johnson and Paulusma asked for the complexity of LI 3-COL in the class of chordal bipartite graphs. In this paper we give a partial answer to this question by showing that LI k-COL is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the LI k-COL in the calss of convex bipartite graphs. Finally we show how our algorithm can be modified to solve the more general LI H-COL problem on convex bipartite graphs.
The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a serie... more The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops on theory and applications of discrete algorithms, graphs and combinatorial structures in the wide sense, organized by either
We consider the following problem: can a certain graph parameter of some given graph be reduced b... more We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pfree graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomialtime solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.
Edge searching is a graph problem that corresponds to cleaning a contaminated graph using the min... more Edge searching is a graph problem that corresponds to cleaning a contaminated graph using the minimum number of searchers. We define fast searching as a variant of this widely studied problem. Fast searching corresponds to an internal monotone search in which every edge is traversed exactly once and searchers are not allowed to jump. We present a linear time algorithm to compute the fast search number of trees. We investigate the fast search number of bipartite graphs. We also propose a general cost function for evaluating search strategies that utilizes both edge searching and fast searching.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:
We introduce the multicolored graph realization problem (MGR). The input to the problem is a colo... more We introduce the multicolored graph realization problem (MGR). The input to the problem is a colored graph (G,φ), i.e., a graph together with a coloring φ on its vertices. We can associate to each colored graph (G,φ) a cluster graph (Gφ) in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices S is multicolored when S has exactly one vertex from each color class. The problem is to decide whether there is a multicolored set S such that, after identifying each vertex in S with its color class, G[S] coincides with Gφ. The MGR problem is related to the well known class of generalized network problems, most of which are NP-hard. For example the generalized Minimun Spanning Tree problem. The MGR problem is a generalization of the multicolored clique problem, which is known to be W [1]-hard when parameterized by the number of colors. Thus MGR remains W [1]-hard, when parameterized by the size of the cluster graph. Thi...
We study the following problem: for given integers d, k and graph G, can we reduce some fixed gra... more We study the following problem: for given integers d, k and graph G, can we reduce some fixed graph parameter π of G by at least d via at most k graph operations from some fixed set S? As parameters we take the chromatic number χ, clique number ω and independence number α, and as operations we choose the edge contraction ec and vertex deletion vd. We determine the complexity of this problem for S = {ec} and S = {vd} and π ∈ {χ, ω, α} for a number of subclasses of perfect graphs. We use these results to determine the complexity of the problem for S = {ec} and S = {vd} and π ∈ {χ, ω, α} restricted to H-free graphs.
In the edge searching problem, searchers move from vertex to vertex in a graph to capture an invi... more In the edge searching problem, searchers move from vertex to vertex in a graph to capture an invisible, fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which, at every move, a new edge of the graph G must be guaranteed to be free of the intruder. That is, once all searchers are placed the graph G is cleared in exactly |E(G)| moves. Such a restriction obviously necessitates a larger number of searchers. We examine this model, and characterize graphs for which 2 or 3 searchers are sufficient. We prove that the corresponding decision problem is NP-complete.
ABSTRACT We consider the following problem: can a certain graph parameter of some given graph be ... more ABSTRACT We consider the following problem: can a certain graph parameter of some given graph be reduced by at most $d$ for some integer~$d$ via at most $k$ edge contractions for some given integer $k$? We consider three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when $d$ is part of the input, this problem is polynomial-time solvable on $P_4$-free graphs and \NP-complete for split graphs. Moreover, as split graphs form a subclass of $P_5$-free graphs, both results together give a complete complexity classification for $P_\ell$-free graphs. On the positive side, for split graphs, we also show that the problem is polynomial-time solvable if $d$ is fixed, that is, not part of the input.
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