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Mordechai Shalom

Publications and source records attributed to Mordechai Shalom.

At least 19 recordsLinked to original sources
arXiv2024

Generation of weighted trees, block trees and block graphs

We present a general framework to generate trees every vertex of which has a non-negative weight and a color. The colors are used to impose certain restrictions on the weight and colors of other vertices. We first extend the enumeration algorithms of unweighted trees given in [19, 20] to generate weighted trees that allow zero weight. We avoid isomorphisms by generalizing the concept of centroids to weighted trees and then using the so-called centroid-rooted canonical weighted trees. We provide a time complexity analysis of unranking algorithms and also show that the output delay complexity of enumeration is linear. The framework can be used to generate graph classes taking advantage of their tree-based decompositions/representations. We demonstrate our framework by generating weighted block trees which are in one-to-one correspondence with connected block graphs. All connected block graphs up to 19 vertices are publicly available at [1].

cs.DM↗
arXiv2020

Defensive Domination in Proper Interval Graphs

$k$-defensive domination, a variant of the classical domination problem on graphs, seeks a minimum cardinality vertex set providing a surjective defense against any attack on vertices of cardinality bounded by a parameter $k$. The problem has been shown to be NP-complete} for fixed $k$; if $k$ is part of the input, the problem is not even in NP. We present efficient algorithms solving this problem on proper interval graphs with $k$ part of the input. The algorithms take advantage of the linear orderings of the end points of the intervals associated with vertices to realize a greedy approach to solution. The first algorithm is based on the interval model and has complexity ${\cal O}(n \cdot k)$ for a graph on $n$ vertices. The second one is an improvement of the first and employs bubble representations of proper interval graph to realize an improved complexity of ${\cal O}(n+ \vert{\cal B}\vert \cdot \log k)$ for a graph represented by $\vert{\cal B}\vert$ bubbles.

cs.DM↗
arXiv2020

On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that we introduce. These new parameters are related to a generalization of the so-called bubble representations of proper interval graphs and mixed unit interval graphs and to clique-width decompositions.

cs.DS↗
arXiv2020

Multicast Communications in Tree Networks with Heterogeneous Capacity Constraints

A widely studied problem in communication networks is that of finding the maximum number of communication requests that can be scheduled concurrently, subject to node and/or link capacity constraints. In this paper, we consider the problem of finding the largest number of multicast communication requests that can be serviced simultaneously by a network of tree topology, subject to heterogeneous capacity constraints. This problem generalizes the following two problems studied in the literature: a) the problem of finding a largest induced $k$-colorable subgraph of a chordal graph, b) the maximum multi-commodity flow problem in tree networks. The problem is already known to be NP-hard and to admit a $c$-approximation ($c \approx 1.58$) in the case of homogeneous capacity constraints. We first show that the problem is much harder to approximate in the heterogeneous case. We then use a generalization of a classical algorithm to obtain an $M$-approximation where $M$ is the maximum number of leaves of the subtrees representing the multicast communications. Surprisingly, the same algorithm, though in various disguises, is used in the literature at least four times to solve related problems (though the analysis is different). The special case of the problem where instances are restricted to unicast communications in a star topology network is known to be polynomial-time solvable. We extend this result and show that the problem can be solved in polynomial time for a set of paths in a tree that share a common vertex.

cs.DS↗
arXiv2019

The Complexity of Subtree Intersection Representation of Chordal Graphs and Linear Time Chordal Graph Generation

It is known that any chordal graph on $n$ vertices can be represented as the intersection of $n$ subtrees in a tree on $n$ nodes. This fact is recently used in [2] to generate random chordal graphs on $n$ vertices by generating $n$ subtrees of a tree on $n$ nodes. It follows that the space (and thus time) complexity of such an algorithm is at least the sum of the sizes of the generated subtrees assuming that a tree is given by a set of nodes. In [2], this complexity was mistakenly claimed to be linear in the number $m$ of edges of the generated chordal graph. This error is corrected in [3] where the space complexity is shown to be $Ω(m n^{1/4})$. The exact complexity of the algorithm is left as an open question. In this paper, we show that the sum of the sizes of $n$ subtrees in a tree on $n$ nodes is $Θ(m\sqrt{n})$. We also show that we can confine ourselves to contraction-minimal subtree intersection representations since they are sufficient to generate every chordal graph. Furthermore, the sum of the sizes of the subtrees in such a representation is at most $2m+n$. We use this result to derive the first linear time random chordal graph generator. Based on contraction-minimal representations, we also derive structural properties of chordal graphs related to their connectivity. In addition to these theoretical results, we conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those in the literature. Our experimental study indicates that the generated graphs do not have a restricted structure and the sizes of maximal cliques are distributed fairly over the range. Furthermore, our algorithm is simple to implement and produces graphs with 10000 vertices and $4 . 10^7$ edges in less than one second on a laptop computer.

cs.DS↗
arXiv2019

Hierarchical b-Matching

A matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a $b$-matching every vertex $v$ has an associated bound $b_v$, and a maximum $b$-matching is a maximum set of edges, such that every vertex $v$ appears in at most $b_v$ of them. We study an extension of this problem, termed {\em Hierarchical b-Matching}. In this extension, the vertices are arranged in a hierarchical manner. At the first level the vertices are partitioned into disjoint subsets, with a given bound for each subset. At the second level the set of these subsets is again partitioned into disjoint subsets, with a given bound for each subset, and so on. In an {\em Hierarchical b-matching} we look for a maximum set of edges, that will obey all bounds (that is, no vertex $v$ participates in more than $b_v$ edges, then all the vertices in one subset do not participate in more that that subset's bound of edges, and so on hierarchically). We propose a polynomial-time algorithm for this new problem, that works for any number of levels of this hierarchical structure.

cs.DS↗
arXiv2019

Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is the smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. % In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to that of finding an $r$-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of $r=2$, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of $r$ and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT algorithm.

cs.CC↗
arXiv2018

Equimatchable Claw-Free Graphs

A graph is equimatchable if all of its maximal matchings have the same size. A graph is claw-free if it does not have a claw as an induced subgraph. In this paper, we provide, to the best of our knowledge, the first characterization of claw-free equimatchable graphs by identifying the equimatchable claw-free graph families. This characterization implies an efficient recognition algorithm.

cs.DM↗
arXiv2018

Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II

Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). We define the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the graph having a vertex for each path in P, and an edge between every pair of vertices representing two paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=ENPT(P), and we say that is a representation of G. Our goal is to characterize the representation of chordless ENPT cycles (holes). To achieve this goal, we first assume that the EPT graph induced by the vertices of an ENPT hole is given. In [2] we introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work, we continue our work by relaxing these three assumptions one by one. We characterize the representations of ENPT holes satisfying (P3) by providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also show that there does not exist a polynomial-time algorithm to solve HamiltonianPairRec, unless P=NP.

cs.DM↗
arXiv2017

Graphs of Edge-Intersecting and Non-Splitting One Bend Paths in a Grid

The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of subclasses that are obtained by restricting the number of bends in the paths. Motivated by this result, in this work we focus on one bend {ENPG} graphs. We show that one bend ENPG graphs are properly included in two bend ENPG graphs. We also show that trees and cycles are one bend ENPG graphs, and characterize the split graphs and co-bipartite graphs that are one bend ENPG. We prove that the recognition problem of one bend ENPG split graphs is NP-complete even in a very restricted subfamily of split graphs. Last we provide a linear time recognition algorithm for one bend ENPG co-bipartite graphs.

cs.DM↗
arXiv2017

Parameterized complexity of finding a spanning tree with minimum reload cost diameter

We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph $G$, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of $G$ of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree $Δ$ of the input graph. We prove that DIAMETER-TREE is para-NP-hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove DIAMETER-TREE to be NP-hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan (2001) proved that the problem can be solved in polynomial time on graphs with $Δ= 3$, and Galbiati (2008) proved that it is NP-hard if $Δ= 4$. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP-hard if $Δ= 3$, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that DIAMETER-TREE is in XP and W[1]-hard parameterized by the treewidth plus $Δ$.

cs.DS↗
arXiv2016

On two extensions of equimatchable graphs

A graph is said to be equimatchable if all its maximal matchings are of the same size. In this work we introduce two extensions of the property of equimatchability by defining two new graph parameters that measure how far a graph is from being equimatchable. The first one, called the matching gap, measures the difference between the sizes of a maximum matching and a minimum maximal matching. The second extension is obtained by introducing the concept of equimatchable sets; a set of vertices in a graph $G$ is said to be equimatchable if all maximal matchings of $G$ saturating the set are of the same size. Noting that $G$ is equimatchable if and only if the empty set is equimatchable, we study the equimatchability defect of the graph, defined as the minimum size of an equimatchable set in it. We develop several inapproximability and parameterized complexity results and algorithms regarding the computation of these two parameters, a characterization of graphs of unit matching gap, exact values of the equimatchability defect of cycles, and sharp bounds for both parameters.

math.CO↗
arXiv2016

Edge Coloring with Minimum Reload/Changeover Costs

In an edge-colored graph, a traversal cost occurs at a vertex along a path when consecutive edges with different colors are traversed. The value of the traversal cost depends only on the colors of the traversed edges. This concept leads to two global cost measures, namely the \emph{reload cost} and the \emph{changeover cost}, that have been studied in the literature and have various applications in telecommunications, transportation networks, and energy distribution networks. Previous work focused on problems with an edge-colored graph being part of the input. In this paper, we formulate and focus on two pairs of problems that aim to find an edge coloring of a graph so as to minimize the reload and changeover costs. The first pair of problems aims to find a proper edge coloring so that the reload/changeover cost of a set of paths is minimized. The second pair of problems aim to find a proper edge coloring and a spanning tree so that the reload/changeover cost is minimized. We present several hardness results as well as polynomial-time solvable special cases.

cs.DS↗
arXiv2016

A Simple Extension of Dirac's Theorem on Hamiltonicity

The classical Dirac theorem asserts that every graph $G$ on $n$ vertices with minimum degree $δ(G) \ge \lceil n/2 \rceil$ is Hamiltonian. The lower bound of $\lceil n/2 \rceil$ on the minimum degree of a graph is tight. In this paper, we extend the classical Dirac theorem to the case where $δ(G) \ge \lfloor n/2 \rfloor $ by identifying the only non-Hamiltonian graph families in this case. We first present a short and simple proof. We then provide an alternative proof that is constructive and self-contained. Consequently, we provide a polynomial-time algorithm that constructs a Hamiltonian cycle, if exists, of a graph $G$ with $δ(G) \ge \lfloor n/2 \rfloor$, or determines that the graph is non-Hamiltonian. Finally, we present a self-contained proof for our algorithm which provides insight into the structure of Hamiltonian cycles when $δ(G) \ge \lfloor n/2 \rfloor$ and is promising for extending the results of this paper to the cases with smaller degree bounds.

cs.DM↗
arXiv2016

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of Gözüpek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in Gözüpek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.

cs.DS↗
arXiv2015

Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Towards Hole Representations-Part I

Given a tree and a set ${\cal P}$ of non-trivial simple paths on it, $VPT({\cal P})$ is the VPT graph (i.e. the vertex intersection graph) of the paths ${\cal P}$ of the tree $T$, and $EPT({\cal P})$ is the EPT graph (i.e. the edge intersection graph) of ${\cal P}$. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their \emph{split vertices} is the set of vertices having degree at least $3$ in their union. A pair of (edge) intersecting paths is termed \emph{non-splitting} if they do not have split vertices (namely if their union is a path). In this work, motivated by an application in all-optical networks, we define the graph $ENPT({\cal P})$ of edge-intersecting non-splitting paths of a tree, termed the ENPT graph, as the (edge) graph having a vertex for each path in ${\cal P}$, and an edge between every pair of paths that are both edge-intersecting and non-splitting. A graph $G$ is an ENPT graph if there is a tree $T$ and a set of paths ${\cal P}$ of $T$ such that $G=ENPT({\cal P})$, and we say that $ $ is a \emph{representation} of $G$. We first show that cycles, trees and complete graphs are ENPT graphs. Our work follows the lines of Golumbic and Jamison's research in which they defined the EPT graph class, and characterized the representations of chordless cycles (holes). It turns out that ENPT holes have a more complex structure than EPT holes. In our analysis, we assume that the EPT graph corresponding to a representation of an ENPT hole is given. We also introduce three assumptions $(P1)$, $(P2)$, $(P3)$ defined on EPT, ENPT pairs of graphs. In this Part I, using the results of Golumbic and Jamison as building blocks, we characterize (a) EPT, ENPT pairs that satisfy $(P1)$, $(P2)$, $(P3)$, and (b) the unique minimal representation of such pairs.

cs.DM↗
arXiv2015

The Maximum Cut Problem in Co-bipartite Chain Graphs

A \emph{co-bipartite chain} graph is a co-bipartite graph in which the neighborhoods of the vertices in each clique can be linearly ordered with respect to inclusion. It is known that the maximum cut problem (MaxCut) is NP-Hard in co-bipartite graphs. We consider MaxCut in co-bipartite chain graphs. We first consider the twin-free case and present an explicit solution. We then show that MaxCut is polynomial time solvable in this graph class.

cs.DS↗