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Register a new userCombinatorial Problems on $H$-graphs
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Added by
Peter Zeman
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Bir{o}, Hujter, and Tuza introduced the concept of $H$-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph $H$. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. We continue the study of these graph classes by considering coloring, clique, and isomorphism problems on $H$-graphs. We show that for any fixed $H$ containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on $H$-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on $H$-graphs. Namely, when $H$ is a cactus the clique problem can be solved in polynomial time. Also, when a graph $G$ has a Helly $H$-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the $k$-clique and list $k$-coloring problems are FPT on $H$-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number.
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Bir{o} et al. (1992) introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a graph $H$. They are related to many classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. We negatively answer the 25-year-old question of Bir{o} et al. which asks if $H$-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. We provide a polynomial-time algorithm recognizing $T$-graphs, for each fixed tree $T$. When $T$ is a star $S_d$ of degree $d$, we have an $O(n^{3.5})$-time algorithm. We give FPT- and XP-time algorithms solving the minimum dominating set problem on $S_d$-graphs and $H$-graphs parametrized by $d$ and the size of $H$, respectively. 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 $H$-graphs are GI-complete and that the clique problem is APX-hard. The clique problem can be solved in polynomial time if $H$ is a cactus graph. When a graph $G$ has a Helly $H$-representation, the clique problem can be solved in polynomial time. We show that both the $k$-clique and the list $k$-coloring problems are solvable in FPT-time on $H$-graphs (parameterized by $k$ and the treewidth of $H$). In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that $H$-graphs have at most $n^{O(|H|)}$ minimal separators which allows us to apply the meta-algorithmic framework of Fomin et al. (2015) to show that for each fixed $t$, finding a maximum induced subgraph of treewidth $t$ can be done in polynomial time. When $H$ is a cactus, we improve the bound to $O(|H|n^2)$.
A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from $G$ to $H$ can be calculated in subexponential time $2^{Oleft(sqrt{tnlog(n)}right)}$ for $n=|V(G)|$ in the class of $P_t$-free graphs $G$. As a corollary, we show that the number of 3-colourings of a $P_t$-free graph $G$ can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of $P_t$-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that $P_t$-free graphs have pathwidth that is linear in their maximum degree.
Different graph generalizations have been recently used in an ad-hoc manner to represent multilayer networks, i.e. systems formed by distinct layers where each layer can be seen as a network. Similar constructions have also been used to represent time-varying networks. We introduce the concept of MultiAspect Graph (MAG) as a graph generalization that we prove to be isomorphic to a directed graph, and also capable of representing all previous generalizations. In our proposal, the set of vertices, layers, time instants, or any other independent features are considered as an aspect of the MAG. For instance, a MAG is able to represent multilayer or time-varying networks, while both concepts can also be combined to represent a multilayer time-varying network and even other higher-order networks. Since the MAG structure admits an arbitrary (finite) number of aspects, it hence introduces a powerful modelling abstraction for networked complex systems. This paper formalizes the concept of MAG and derives theoretical results useful in the analysis of complex networked systems modelled using the proposed MAG abstraction. We also present an overview of the MAG applicability.
A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as $L(1,1)$-Labelling). A classical complexity result on Colouring is a well-known dichotomy for $H$-free graphs (a graph is $H$-free if it does not contain $H$ as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on $H$-free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours $k$ is fixed, that is, not part of the input. From our study it follows that for fixed $k$ the three problems behave in the same way, but this is no longer true if $k$ is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.
In this paper we present an algorithmic framework for solving a class of combinatorial optimization problems on graphs with bounded pathwidth. The problems are NP-hard in general, but solvable in linear time on this type of graphs. The problems are relevant for assessing network reliability and improving the networks performance and fault tolerance. The main technique considered in this paper is dynamic programming.
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