We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or... more We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty (or, equivalently, k = 0), the stated problem coincides with the classical path cover problem. The kPC problem generalizes some path cover related problems, such as the 1HP and 2HP problems, which have been proved to be NP-complete. Note that, the complexity status of both 1HP and 2HP problems on interval graphs remains an open question [9]. In this paper, we show that the kPC problem can be solved in linear time on the class of proper interval graphs, that is, in O(n + m) time on a proper interval graph on n vertices and m edges. The proposed algorithm is simple, requires linear space, and also enables us to solve the 1HP and 2HP problems, on proper interval graphs within the same time and space complexity.
In this paper we study a generalization of the path cover problem, namely, the 2-terminal-set pat... more In this paper we study a generalization of the path cover problem, namely, the 2-terminal-set path cover problem, or 2TPC for short. Given a graph G and two disjoint subsets T 1 T^ 1 and T 2 T^ 2 of V (G), a 2-terminal-set path cover of G with respect to T 1 T^ 1 and T 2 T^ 2 ...
Extending previous NP-completeness results for the harmonious coloring problem and the pair-compl... more Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Comput. Sci. 290 (2003) 2147-2155], where he left the problem open for the class of convex graphs, we prove that the kpath partition problem is NP-complete on convex graphs. Moreover, we study the complexity of these problems on two well-known subclasses of chordal graphs namely quasi-threshold and threshold graphs. Based on the work of Bodlaender [H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform. Process. Lett. 31 (1989) 135-138], we show NPcompleteness results for the pair-complete coloring and harmonious coloring problems on quasi-threshold graphs. Concerning the k-path partition problem, we prove that it is also NP-complete on this class of graphs. It is known that both the harmonious coloring problem and the k-path partition problem are polynomially solvable on threshold graphs. We show that the pair-complete coloring problem is also polynomially solvable on threshold graphs by describing a linear-time algorithm.
In this paper, we study a variant of the path cover problem, namely, the k-fixed-endpoint path co... more In this paper, we study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a kfixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The k-fixed-endpoint path cover problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty, that is, k = 0, the stated problem coincides with the classical path cover problem. We show that the k-fixed-endpoint path cover problem can be solved in linear time on the class of cographs. More precisely, we first establish a lower bound on the size of a minimum k-fixed-endpoint path cover of a cograph and prove structural properties for the paths of such a path cover. Then, based on these properties, we describe an algorithm which, for a cograph G on n vertices and m edges, computes a minimum k-fixed-endpoint path cover of G in linear time, that is, in O(n + m) time. The proposed algorithm is simple, requires linear space, and also enables us to solve some path cover related problems, such as the 1HP and 2HP, on cographs within the same time and space complexity.
Journal of Parallel and Distributed Computing, 2007
Nakano et al. in [20] presented a time-and work-optimal algorithm for finding the smallest number... more Nakano et al. in [20] presented a time-and work-optimal algorithm for finding the smallest number of vertex-disjoint paths that cover the vertices of a cograph and left open the problem of applying their technique into other classes of graphs. Motivated by this issue we generalize their technique and apply it to the class of P 4-sparse graphs, which forms a proper superclass of cographs. We show that the path cover problem on P 4-sparse graphs can also be optimally solved. More precisely, given a P 4-sparse graph G on n vertices and its modular decomposition tree, we describe an optimal parallel algorithm which returns a minimum path cover of G in O(log n) time using O(n/ log n) processors on the EREW PRAM model. Our results generalize previous results and extend the family of perfect graphs admitting optimal solutions for the path cover problem.
This work focuses on the complexity status of path cover and coloring problems on classes of perf... more This work focuses on the complexity status of path cover and coloring problems on classes of perfect graphs. In particular, it deals with designing and analyzing graph algorithms for these problems or, for the cases where this is not possible, with providing NP-completeness results. Most of our results have already been published. We next describe the variations of the path cover and coloring problems that we study in this work. […]Αυτή η εργασία επικεντρώνεται στη μελέτη της πολυπλοκότητας προβλημάτων επικάλυψης με μονοπάτια και χρωματισμού σε κλάσεις τέλειων γραφημάτων. Συγκεκριμένα, ασχολούμαστε με την σχεδίαση και ανάλυση αλγορίθμων για αυτά τα προβλήματα ή, για τις περιπτώσεις που αυτό δεν είναι δυνατό, δίνουμε αποτελέσματα NP-πληρότητας. Τα περισσότερα αποτελέσματα αυτής της εργασίας έχουν ήδη δημοσιευτεί. Στην συνέχεια περιγράφουμε επεκτάσεις προβλημάτων επικάλυψης και χρωματισμού που μας απασχολούν στη συγκεκριμένη εργασία. […
We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover proble... more We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph $G$ and a subset $\mathcal{T}$ of $k$ vertices of $V(G)$, a $k$-fixed-endpoint path cover of $G$ with respect to $\mathcal{T}$ is a set of vertex-disjoint paths $\mathcal{P}$ that covers the vertices of $G$ such that the $k$ vertices of $\mathcal{T}$ are all endpoints of the paths in $\mathcal{P}$. The kPC problem is to find a $k$-fixed-endpoint path cover of $G$ of minimum cardinality; note that, if $\mathcal{T}$ is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke, where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can...
We study a variant of the path cover problem, namely, the k-flxed-endpoint path cover problem, or... more We study a variant of the path cover problem, namely, the k-flxed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-flxed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k
In this paper, we prove that the harmonious coloring problem is NP-complete for connected interva... more In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.
This paper describes efficient data structures, namely the Indexed P-tree, Block P-tree, and Inde... more This paper describes efficient data structures, namely the Indexed P-tree, Block P-tree, and Indexed-Block P-tree (or IP-tree, BP-tree, and IBP-tree, respectively, for short), for maintaining future events in a general purpose discrete event simulation system, and studies the performance of their event set algorithms under the event horizon principle. For comparison reasons, some well-known event set algorithms were also selected and studied; that is, the Dynamic-heap and the P-tree algorithms. To gain insight into the performance of the proposed event set algorithms and allow comparisons with the other selected algorithms, they are tested under a wide variety of conditions in an experimental way. The time needed for the execution of the Hold operation is taken as the measure for estimating the average time complexity of the algorithms. The experimental results show that the BP-tree algorithm and the IBP-tree algorithm behave very well with all the sizes of the event set and their performance is almost independent from the stochastic distributions.
We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or... more We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty (or, equivalently, k = 0), the stated problem coincides with the classical path cover problem. The kPC problem generalizes some path cover related problems, such as the 1HP and 2HP problems, which have been proved to be NP-complete. Note that, the complexity status of both 1HP and 2HP problems on interval graphs remains an open question [9]. In this paper, we show that the kPC problem can be solved in linear time on the class of proper interval graphs, that is, in O(n + m) time on a proper interval graph on n vertices and m edges. The proposed algorithm is simple, requires linear space, and also enables us to solve the 1HP and 2HP problems, on proper interval graphs within the same time and space complexity.
In this paper we study a generalization of the path cover problem, namely, the 2-terminal-set pat... more In this paper we study a generalization of the path cover problem, namely, the 2-terminal-set path cover problem, or 2TPC for short. Given a graph G and two disjoint subsets T 1 T^ 1 and T 2 T^ 2 of V (G), a 2-terminal-set path cover of G with respect to T 1 T^ 1 and T 2 T^ 2 ...
Extending previous NP-completeness results for the harmonious coloring problem and the pair-compl... more Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Comput. Sci. 290 (2003) 2147-2155], where he left the problem open for the class of convex graphs, we prove that the kpath partition problem is NP-complete on convex graphs. Moreover, we study the complexity of these problems on two well-known subclasses of chordal graphs namely quasi-threshold and threshold graphs. Based on the work of Bodlaender [H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform. Process. Lett. 31 (1989) 135-138], we show NPcompleteness results for the pair-complete coloring and harmonious coloring problems on quasi-threshold graphs. Concerning the k-path partition problem, we prove that it is also NP-complete on this class of graphs. It is known that both the harmonious coloring problem and the k-path partition problem are polynomially solvable on threshold graphs. We show that the pair-complete coloring problem is also polynomially solvable on threshold graphs by describing a linear-time algorithm.
In this paper, we study a variant of the path cover problem, namely, the k-fixed-endpoint path co... more In this paper, we study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a kfixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The k-fixed-endpoint path cover problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty, that is, k = 0, the stated problem coincides with the classical path cover problem. We show that the k-fixed-endpoint path cover problem can be solved in linear time on the class of cographs. More precisely, we first establish a lower bound on the size of a minimum k-fixed-endpoint path cover of a cograph and prove structural properties for the paths of such a path cover. Then, based on these properties, we describe an algorithm which, for a cograph G on n vertices and m edges, computes a minimum k-fixed-endpoint path cover of G in linear time, that is, in O(n + m) time. The proposed algorithm is simple, requires linear space, and also enables us to solve some path cover related problems, such as the 1HP and 2HP, on cographs within the same time and space complexity.
Journal of Parallel and Distributed Computing, 2007
Nakano et al. in [20] presented a time-and work-optimal algorithm for finding the smallest number... more Nakano et al. in [20] presented a time-and work-optimal algorithm for finding the smallest number of vertex-disjoint paths that cover the vertices of a cograph and left open the problem of applying their technique into other classes of graphs. Motivated by this issue we generalize their technique and apply it to the class of P 4-sparse graphs, which forms a proper superclass of cographs. We show that the path cover problem on P 4-sparse graphs can also be optimally solved. More precisely, given a P 4-sparse graph G on n vertices and its modular decomposition tree, we describe an optimal parallel algorithm which returns a minimum path cover of G in O(log n) time using O(n/ log n) processors on the EREW PRAM model. Our results generalize previous results and extend the family of perfect graphs admitting optimal solutions for the path cover problem.
This work focuses on the complexity status of path cover and coloring problems on classes of perf... more This work focuses on the complexity status of path cover and coloring problems on classes of perfect graphs. In particular, it deals with designing and analyzing graph algorithms for these problems or, for the cases where this is not possible, with providing NP-completeness results. Most of our results have already been published. We next describe the variations of the path cover and coloring problems that we study in this work. […]Αυτή η εργασία επικεντρώνεται στη μελέτη της πολυπλοκότητας προβλημάτων επικάλυψης με μονοπάτια και χρωματισμού σε κλάσεις τέλειων γραφημάτων. Συγκεκριμένα, ασχολούμαστε με την σχεδίαση και ανάλυση αλγορίθμων για αυτά τα προβλήματα ή, για τις περιπτώσεις που αυτό δεν είναι δυνατό, δίνουμε αποτελέσματα NP-πληρότητας. Τα περισσότερα αποτελέσματα αυτής της εργασίας έχουν ήδη δημοσιευτεί. Στην συνέχεια περιγράφουμε επεκτάσεις προβλημάτων επικάλυψης και χρωματισμού που μας απασχολούν στη συγκεκριμένη εργασία. […
We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover proble... more We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph $G$ and a subset $\mathcal{T}$ of $k$ vertices of $V(G)$, a $k$-fixed-endpoint path cover of $G$ with respect to $\mathcal{T}$ is a set of vertex-disjoint paths $\mathcal{P}$ that covers the vertices of $G$ such that the $k$ vertices of $\mathcal{T}$ are all endpoints of the paths in $\mathcal{P}$. The kPC problem is to find a $k$-fixed-endpoint path cover of $G$ of minimum cardinality; note that, if $\mathcal{T}$ is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke, where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can...
We study a variant of the path cover problem, namely, the k-flxed-endpoint path cover problem, or... more We study a variant of the path cover problem, namely, the k-flxed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-flxed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k
In this paper, we prove that the harmonious coloring problem is NP-complete for connected interva... more In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.
This paper describes efficient data structures, namely the Indexed P-tree, Block P-tree, and Inde... more This paper describes efficient data structures, namely the Indexed P-tree, Block P-tree, and Indexed-Block P-tree (or IP-tree, BP-tree, and IBP-tree, respectively, for short), for maintaining future events in a general purpose discrete event simulation system, and studies the performance of their event set algorithms under the event horizon principle. For comparison reasons, some well-known event set algorithms were also selected and studied; that is, the Dynamic-heap and the P-tree algorithms. To gain insight into the performance of the proposed event set algorithms and allow comparisons with the other selected algorithms, they are tested under a wide variety of conditions in an experimental way. The time needed for the execution of the Hold operation is taken as the measure for estimating the average time complexity of the algorithms. The experimental results show that the BP-tree algorithm and the IBP-tree algorithm behave very well with all the sizes of the event set and their performance is almost independent from the stochastic distributions.
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