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Papers by ton kloks

Research paper thumbnail of Folding graphs

arXiv (Cornell University), 2012

Let G be a graph. Consider two nonadjacent vertices x and y that have a common neighbor. Folding ... more Let G be a graph. Consider two nonadjacent vertices x and y that have a common neighbor. Folding G with respect to x and y is the operation which identifies x and y. After a maximal series of foldings the graph is a disjoint union of cliques. The minimal clique number that can appear after a maximal series of foldings is equal to the chromatic number of G. In this paper we consider the problem to determine the maximal clique number which can appear after a maximal series of foldings. We denote this number as Sigma(G) and we call it the max-folding number. We show that the problem is NP-complete, even when restricted to classes such as trivially perfect graphs, cobipartite graphs and planar graphs. We show that the max-folding number of trees is two.

Research paper thumbnail of Complexity of paired domination in AT-free and planar graphs

Theoretical Computer Science, Sep 1, 2022

For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not... more For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

Research paper thumbnail of Triangulating 3-colored graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Complexity of Paired Domination in AT-free and Planar Graphs

arXiv (Cornell University), Dec 10, 2021

For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not... more For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

Research paper thumbnail of Tracking maximum ascending subsequences in sequences of partially ordered data

UU BETA ICS Departement Informatica eBooks, May 1, 2017

We consider scenarios in which long sequences of data are analyzed and subsequences must be trace... more We consider scenarios in which long sequences of data are analyzed and subsequences must be traced that are monotone and maximum, according to some measure. A classical example is the online Longest Increasing Subsequence Problem for numeric and alphanumeric data. We extend the problem in two ways: (a) we allow data from any partially ordered set, and (b) we maximize subsequences using much more general measures than just length or weight. Let P be a poset of finite width w, and let δ be any data sequence over P. We show that the measure of the maximum monotone subsequences in δ can be maintained in at most O(w log min(n w , Dn)) time and O(min(n, wDn)) memory when the n-th data item is processed, where Dn is the 'depth' of the measure at position n (n ≥ 1). The result generalizes all earlier O(log n) time-per-input results for the corresponding longest or heaviest increasing subsequence problems.

Research paper thumbnail of Pathwidth of pathwidth-bounded graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Gray Codes for AT-Free Orders

arXiv (Cornell University), Jun 20, 2017

AT-free graphs are characterized by vertex elimination orders. We show that these AT-free orders ... more AT-free graphs are characterized by vertex elimination orders. We show that these AT-free orders of a graph can be generated in constant amortized time. ⋆ A preliminary version "Gray codes for AT-free orders via antimatroids" was presented in the 26th International Workshop on Combinatorial Algorithms (IWOCA 2015), Verona, Italy.

Research paper thumbnail of Treewidth of treewidth-bounded graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Recognizing treewidth-bounded graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Approximating treewidth and pathwidth of a graph

Research paper thumbnail of On the Grundy number of Cameron graphs

Electronic Notes in Discrete Mathematics, Dec 1, 2017

The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of ... more The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of the graph. The class of Cameron graphs is the Seidel switching class of cographs. In this paper we show that the Grundy number is computable in polynomial time for Cameron graphs.

Research paper thumbnail of Recent Trends

Research paper thumbnail of Problem Formulations

Research paper thumbnail of A Generalization of AT-free Graphs and a Generic Algorithm for Solving Treewidth, Minimum Fill-In and Vertex Ranking

Springer eBooks, 1998

A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A, the set A ... more A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A, the set A \ {a} is contained in one component of G − N [a]. An asteroidal set of cardinality three is called asteroidal triple and graphs without an asteroidal triple are called AT-free. The maximum cardinality of an asteroidal set of G, denoted by an(G), is said to be the asteroidal number of G. We present a scheme for designing algorithms for triangulation problems on graphs. As a consequence, we obtain algorithms to compute graph parameters such as treewidth, minimum fill-in and vertex ranking number. The running time of these algorithms is a polynomial (of degree asteroidal number plus a small constant) in the number of vertices and the number of minimal separators of the input graph.

Research paper thumbnail of Complexity of Paired Domination in At-Free and Planar Graphs

Social Science Research Network, 2022

For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not... more For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

Research paper thumbnail of Treewidth and pathwidth of permutation graphs

Research paper thumbnail of On the strong chromatic index and induced matching of tree-cographs, permutation graphs and chordal bipartite graphs

arXiv (Cornell University), Oct 17, 2011

We show that there exist linear-time algorithms that compute the strong chromatic index and a max... more We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of treecographs when the decomposition tree is a part of the input. We also show that there exist efficient algorithms for the strong chromatic index of (bipartite) permutation graphs and of chordal bipartite graphs.

Research paper thumbnail of On the Grundy number of Cameron graphs

arXiv (Cornell University), Apr 25, 2016

The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of ... more The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of the graph. The class of Cameron graphs is the Seidel switching class of cographs. In this paper we show that the Grundy number is computable in polynomial time for Cameron graphs.

Research paper thumbnail of A simple linear time algorithm for triangulating three-colored graphs

Research paper thumbnail of A Class of Graphs Containing the Polar Spaces

European Journal of Combinatorics, Apr 1, 1986

We consider finite graphs with the property that there exists a constant e such that for every ma... more We consider finite graphs with the property that there exists a constant e such that for every maximal clique M and vertex x not in M, x is adjacent to exactly e vertices in M. It is shown that these graphs have a highly geometric structure which in many ways resembles that of the polar spaces.

Research paper thumbnail of Folding graphs

arXiv (Cornell University), 2012

Let G be a graph. Consider two nonadjacent vertices x and y that have a common neighbor. Folding ... more Let G be a graph. Consider two nonadjacent vertices x and y that have a common neighbor. Folding G with respect to x and y is the operation which identifies x and y. After a maximal series of foldings the graph is a disjoint union of cliques. The minimal clique number that can appear after a maximal series of foldings is equal to the chromatic number of G. In this paper we consider the problem to determine the maximal clique number which can appear after a maximal series of foldings. We denote this number as Sigma(G) and we call it the max-folding number. We show that the problem is NP-complete, even when restricted to classes such as trivially perfect graphs, cobipartite graphs and planar graphs. We show that the max-folding number of trees is two.

Research paper thumbnail of Complexity of paired domination in AT-free and planar graphs

Theoretical Computer Science, Sep 1, 2022

For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not... more For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

Research paper thumbnail of Triangulating 3-colored graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Complexity of Paired Domination in AT-free and Planar Graphs

arXiv (Cornell University), Dec 10, 2021

For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not... more For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

Research paper thumbnail of Tracking maximum ascending subsequences in sequences of partially ordered data

UU BETA ICS Departement Informatica eBooks, May 1, 2017

We consider scenarios in which long sequences of data are analyzed and subsequences must be trace... more We consider scenarios in which long sequences of data are analyzed and subsequences must be traced that are monotone and maximum, according to some measure. A classical example is the online Longest Increasing Subsequence Problem for numeric and alphanumeric data. We extend the problem in two ways: (a) we allow data from any partially ordered set, and (b) we maximize subsequences using much more general measures than just length or weight. Let P be a poset of finite width w, and let δ be any data sequence over P. We show that the measure of the maximum monotone subsequences in δ can be maintained in at most O(w log min(n w , Dn)) time and O(min(n, wDn)) memory when the n-th data item is processed, where Dn is the 'depth' of the measure at position n (n ≥ 1). The result generalizes all earlier O(log n) time-per-input results for the corresponding longest or heaviest increasing subsequence problems.

Research paper thumbnail of Pathwidth of pathwidth-bounded graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Gray Codes for AT-Free Orders

arXiv (Cornell University), Jun 20, 2017

AT-free graphs are characterized by vertex elimination orders. We show that these AT-free orders ... more AT-free graphs are characterized by vertex elimination orders. We show that these AT-free orders of a graph can be generated in constant amortized time. ⋆ A preliminary version "Gray codes for AT-free orders via antimatroids" was presented in the 26th International Workshop on Combinatorial Algorithms (IWOCA 2015), Verona, Italy.

Research paper thumbnail of Treewidth of treewidth-bounded graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Recognizing treewidth-bounded graphs

Springer eBooks, Mar 6, 2006

Research paper thumbnail of Approximating treewidth and pathwidth of a graph

Research paper thumbnail of On the Grundy number of Cameron graphs

Electronic Notes in Discrete Mathematics, Dec 1, 2017

The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of ... more The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of the graph. The class of Cameron graphs is the Seidel switching class of cographs. In this paper we show that the Grundy number is computable in polynomial time for Cameron graphs.

Research paper thumbnail of Recent Trends

Research paper thumbnail of Problem Formulations

Research paper thumbnail of A Generalization of AT-free Graphs and a Generic Algorithm for Solving Treewidth, Minimum Fill-In and Vertex Ranking

Springer eBooks, 1998

A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A, the set A ... more A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A, the set A \ {a} is contained in one component of G − N [a]. An asteroidal set of cardinality three is called asteroidal triple and graphs without an asteroidal triple are called AT-free. The maximum cardinality of an asteroidal set of G, denoted by an(G), is said to be the asteroidal number of G. We present a scheme for designing algorithms for triangulation problems on graphs. As a consequence, we obtain algorithms to compute graph parameters such as treewidth, minimum fill-in and vertex ranking number. The running time of these algorithms is a polynomial (of degree asteroidal number plus a small constant) in the number of vertices and the number of minimal separators of the input graph.

Research paper thumbnail of Complexity of Paired Domination in At-Free and Planar Graphs

Social Science Research Network, 2022

For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not... more For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

Research paper thumbnail of Treewidth and pathwidth of permutation graphs

Research paper thumbnail of On the strong chromatic index and induced matching of tree-cographs, permutation graphs and chordal bipartite graphs

arXiv (Cornell University), Oct 17, 2011

We show that there exist linear-time algorithms that compute the strong chromatic index and a max... more We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of treecographs when the decomposition tree is a part of the input. We also show that there exist efficient algorithms for the strong chromatic index of (bipartite) permutation graphs and of chordal bipartite graphs.

Research paper thumbnail of On the Grundy number of Cameron graphs

arXiv (Cornell University), Apr 25, 2016

The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of ... more The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of the graph. The class of Cameron graphs is the Seidel switching class of cographs. In this paper we show that the Grundy number is computable in polynomial time for Cameron graphs.

Research paper thumbnail of A simple linear time algorithm for triangulating three-colored graphs

Research paper thumbnail of A Class of Graphs Containing the Polar Spaces

European Journal of Combinatorics, Apr 1, 1986

We consider finite graphs with the property that there exists a constant e such that for every ma... more We consider finite graphs with the property that there exists a constant e such that for every maximal clique M and vertex x not in M, x is adjacent to exactly e vertices in M. It is shown that these graphs have a highly geometric structure which in many ways resembles that of the polar spaces.