Combinatorial Problems on H-graphs

Chaplick, Steven; Zeman, Peter

doi:10.1016/j.endm.2017.06.042

Cited by 20 publications

(22 citation statements)

References 22 publications

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“…Recent developments. Subsequent to the publication of the two conference articles [11,12] (which this paper includes and extends), there have already been further developments regarding H-graphs. For example, two manuscripts have recently appeared on ArXiv.org [18,32].…”

Section: Openmentioning

confidence: 99%

On H-Topological Intersection Graphs

Chaplick

Töpfer

Voborník

et al. 2017

Graph-Theoretic Concepts in Computer Science

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Biró, Hujter, and Tuza (1992) introduced the concept of H-graphs, intersection graphs of connected subregions of a graph H thought of as a one-dimensional topological space. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying H-graphs from the point of view of computational problems that are fundamental in theoretical computer science: recognition, graph isomorphism, dominating set, clique, and colorability.Surprisingly, we negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is NP-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T -graphs, for each fixed tree T . For the special case when T is a star S d of degree d, we have an O(n 3.5 )-time algorithm.We give FPTand XP-time algorithms solving the minimum dominating set problem on S dgraphs and H-graphs parametrized by d and the size of H, respectively. As a byproduct, the algorithm for H-graphs adapts to an XP-time algorithm for the independent set and the independent dominating set problems on H-graphs.If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph G has a Helly H-representation, the clique problem can be solved in polynomial time.Further, we show that both the k-clique and the list k-coloring problems are solvable in FPTtime on H-graphs (parameterized by k and the treewidth of H). In fact, these results apply to classes graphs of graphs with treewidth bounded by a function of the clique number.Finally, we observe that H-graphs have at most n O( H ) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to O( H n 2 ).

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Section: Openmentioning

confidence: 99%

On H-Topological Intersection Graphs

Chaplick

Töpfer

Voborník

et al. 2017

Graph-Theoretic Concepts in Computer Science

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“…It is a natural question whether at least some of these algorithmic results can be extended to more general classes of intersection graphs. Chaplick et al [CTVZ17] and Chaplick and Zeman [CZ17] initiated the systematic study of algorithmic properties of H-graphs. They showed that a number of fundamental optimization problems like Maximum Independent Set and Minimum Dominating Set are solvable in polynomial time on H-graphs for any fixed H. Most of the algorithms developed on H-graphs in [CTVZ17,CZ17] run in time n f (H) , where n is the number of vertices in the input graph and f is some function.…”

Section: Introductionmentioning

confidence: 99%

On the Tractability of Optimization Problems on H-Graphs

2020

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For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. H-graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This class was introduced in the early 1990s by Bíró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Maximum Clique, Maximum Independent Set, or Minimum Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions.First we show that for every fixed H, the class of H-graphs is of logarithmicallybounded boolean-width (via mim-width). Pipelined with the plethora of known algorithms on graphs of bounded boolean-width, this describes a large class of problems solvable in polynomial time on H-graphs. We also observe that H-graphs are graphs with polynomially many minimal separators. Combined with the work of Fomin, Todinca and Villanger on algorithmic properties of such classes of graphs, this identify another wide class of problems solvable in polynomial time on H-graphs.The most fundamental optimization problems among the problems solvable in polynomial time on H-graphs are Maximum Clique, Maximum Independent Set, and Minimum Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Maximum Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, then Minimum Dominating Set is fixed-parameter tractable (FPT) parameterized by the size of H.For Maximum Clique we show that it admits a polynomial kernel parameterized by H and the solution size.

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“…For example, what can be said about Clique Cover? Recall that several other classes of optimization problems have already been considered on H-graphs [12,13,21].…”

Section: Discussionmentioning

confidence: 99%

“…Various aspects of proper interval and proper circular-arc representations have been well studied, and our goal is again to study how these carry to general proper H-graphs. Clearly, all positive algorithmic results obtained for H-graphs in [12,13,21] are valid for proper H-graphs, but since we consider a more restricted graph class, we hope that the tractability area can be expanded.…”

mentioning

confidence: 99%

Kernelization of Graph Hamiltonicity: Proper H-Graphs

Chaplick

Fomin

Golovach

et al. 2019

Lecture Notes in Computer Science

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We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choice of parameterization is strongly influenced by the work of Bir\' o, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi-)graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results. Path Cover admits a kernel of size \scrO (\| H\| 8 ), where \| H\| is the size of graph H. In other words, we design an algorithm that for an n-vertex graph G and integer k \geq 1, in time polynomial in n and \| H\| , outputs a graph G \prime of size \scrO (\| H\| 8 ) and k \prime \leq | V (G \prime )| such that the vertex set of G is coverable by k vertex-disjoint paths if and only if the vertex set of G \prime is coverable by k \prime vertex-disjoint paths. Hamiltonian Cycle admits a kernel of size \scrO (\| H\| 8 ). Cycle Cover admits a polynomial kernel. We prove it by providing a compression of size \scrO (\| H\| 10 ) into another NP-complete problem, namely, Prize Collecting Cycle Cover, that is, we design an algorithm that, in time polynomial in n and \| H\| , outputs an equivalent instance of Prize Collecting Cycle Cover of size \scrO (\| H\| 10 ). In all our algorithms we assume that a proper H-decomposition is given as a part of the input.

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Combinatorial Problems on H-graphs

Cited by 20 publications

References 22 publications

On H-Topological Intersection Graphs

On H-Topological Intersection Graphs

On the Tractability of Optimization Problems on H-Graphs

Kernelization of Graph Hamiltonicity: Proper H-Graphs

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