P. Hell | Simon Fraser University (original) (raw)

Papers by P. Hell

SIAM Journal on Discrete Mathematics, 2020

We unify several seemingly different graph and digraph classes under one umbrella. These classes ... more We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affect whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. 1

arXiv (Cornell University), Aug 11, 2016

In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated persp... more In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction problems; on the other hand, they are a very broad (in a certain sense the broadest reasonable) generalization of interval graphs. The class of bi-arc digraphs is precisely the class of digraphs that admit conservative semilattice polymorphisms. There is much interest in understanding structures that admit particular types of polymorphisms, and especially in their recognition algorithms. (Such recognition problems are usually referred to as "metaproblems".) It turns out that the class of bi-arc digraphs also precisely describes the class of digraphs that admit certain other kinds of conservative polymorphisms (cyclic, and totally symmetric, polymorphisms of all arities). Thus solving the recognition problem for bi-arc digraphs solves the metaproblem for digraphs for several types of conservative polymorphisms. The complexity of the recognition problem for digraphs with conservative semilattice polymorphisms was an open problem, while it was known that the problem is NP-complete for certain more complex relational structures. We complement our result by providing a complete dichotomy classification of which general relational structures have polynomial or NP-complete recognition problems for the existence of conservative semilattice polymorphisms. The class of bi-arc digraphs also generalizes the class of interval graphs; in fact it reduces to the class of interval graphs for digraphs that are symmetric and reflexive. It is much broader than interval graphs and includes other generalizations of interval graphs such as co-threshold tolerance graphs and adjusted interval digraphs. Yet, it is still a reasonable extension of interval graphs, in the sense that it keeps much of the appeal of interval graphs (as we show in this paper). Our main result is a forbidden obstruction characterization of, and a polynomial recognition for, the class of bi-arc digraphs. This is accomplished by a detailed analysis of possible structures in the space of ordered pairs of vertices of a digraph. * This version of the paper is different from the previous arXiv versions in several ways. Apart from continuing to improve the overall presentation (in particular the algorithm and Section 9), in this version we also discuss the complexity of the recognition problem for higher arity structures that admit a conservative semilattice polymorphism; we obtain a full dichotomy classification of this problem for general relational structures.

arXiv (Cornell University), Jan 23, 2011

The Dichotomy Conjecture for constraint satisfaction problems has been verified for conservative ... more The Dichotomy Conjecture for constraint satisfaction problems has been verified for conservative problems (or, equivalently, for list homomorphism problems) by Andrei Bulatov. An earlier case of this dichotomy, for list homomorphisms to undirected graphs, came with an elegant structural distinction between the tractable and intractable cases. Such structural characterization is absent in Bulatov's classification, and Bulatov asked whether one can be found. We provide an answer in the case of digraphs; the technique will apply in a broader context. The key concept we introduce is that of a digraph asteroidal triple (DAT). The dichotomy then takes the following form. If a digraph H has a DAT, then the list homomorphism problem for H is NP-complete; and a DAT-free digraph H has a polynomial time solvable list homomorphism problem. DAT-free graphs can be recognized in polynomial time.

arXiv (Cornell University), Jul 20, 2020

Weighted Szeged index is a recently introduced extension of the well-known Szeged index. In this ... more Weighted Szeged index is a recently introduced extension of the well-known Szeged index. In this paper, we present a new tool to analyze and characterize minimum weighted Szeged index trees. We exhibit the best trees with up to 81 vertices and use this information, together with our results, to propose various conjectures on the structure of minimum weighted Szeged index trees.

Discrete Applied Mathematics, 2018

A graph is a cograph if it is P 4-free. A k-polar partition of a graph G is a partition of the se... more A graph is a cograph if it is P 4-free. A k-polar partition of a graph G is a partition of the set of vertices of G into parts A and B such that the subgraph induced by A is a complete multipartite graph with at most k parts, and the subgraph induced by B is a disjoint union of at most k cliques with no other edges. It is known that k-polar cographs can be characterized by a finite family of forbidden induced subgraphs, for any fixed k. A concrete family of such forbidden induced subgraphs is known for k = 1, since 1-polar graphs are precisely split graphs. For larger k such families are not known, and Ekim, Mahadev, and de Werra explicitely asked for the family for k = 2. In this paper we provide such a family, and show that the graphs can be obtained from four basic graphs by a natural operation that preserves 2-polarity and also preserves the condition of being a cograph. We do not know such an operation for k > 2, nevertheless we believe that the results and methods discussed here will also be useful for higher k.

Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured t... more Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. Goodey proved the conjecture for the intersection of the two classes. We examine these classes from the point of view of distance-two colorings. A distance-two r-coloring of a graph G is an assignment of r colors to the vertices of G so that any two vertices at distance at most two have different colors. Note that a cubic graph needs at least four colors. The distance-two four-coloring problem for cubic planar graphs is known to be NP-complete. We claim the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which we call type-one Barnette graphs, since they are the first class identified by Barnette. By contrast, we claim the problem is polynomial for cubic plane graphs with face sizes 3, 4, 5, or 6, which we call type-two Barnette graphs, because of their relation to Barnette’s second conjecture. We call Goodey graphs those ...

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arXiv (Cornell University), Sep 8, 2019

We define strongly chordal digraphs, which generalize strongly chordal graphs, and chordal bipart... more We define strongly chordal digraphs, which generalize strongly chordal graphs, and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0, 1 matrices that admit a simultaneous row and column permutation avoiding the Γ matrix. In general, it is not clear if these digraphs can be recognized in polynomial time, and we focus on symmetric digraphs (i.e., graphs with possible loops), tournaments with possible loops, and balanced digraphs. In each of these cases we give a polynomial-time recognition algorithm and a forbidden induced subgraph characterization. We also discuss an algorithm for minimum general dominating set in strongly chordal graphs with possible loops, extending and unifying similar algorithms for strongly chordal graphs and chordal bipartite graphs.

Annals of Discrete Mathematics, 1978

... Alspach Pavol Hell Donald J. Miller Page 9. CONTENTS Introduction Contents DG CORNEIL and RA ... more ... Alspach Pavol Hell Donald J. Miller Page 9. CONTENTS Introduction Contents DG CORNEIL and RA MATHON, Algorithmic techniques for the generation and analysis of strongly regular graphs and other com-binatorial conﬁgurations G. DANARAJ and V. KLEE, Which spheres ...

European Journal of Combinatorics, 2017

We investigate the complexity of generalizations of colourings (acyclic colourings, (k,)colouring... more We investigate the complexity of generalizations of colourings (acyclic colourings, (k,)colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.

Electronic Notes in Discrete Mathematics, Dec 1, 2009

In this work we investigate the algorithmic complexity of computing a minimum C k-transversal, i.... more In this work we investigate the algorithmic complexity of computing a minimum C k-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k ≥ 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k ≤ 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k ≥ 3. We also discuss polynomial-time approximation algorithms for computing C3-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.

Discrete Applied Mathematics

This special issue contains most of the invited addresses presented during the Workshop on Comput... more This special issue contains most of the invited addresses presented during the Workshop on Computational Combinatorics.

DIMACS Series in Discrete Mathematics and Theoretical Computer Science

Dedicated to the memory of G. A. Dirac The bandwidth (bandsize) of a graph G is the minimum, over... more Dedicated to the memory of G. A. Dirac The bandwidth (bandsize) of a graph G is the minimum, over all bijections p : V(G)-) {1,2,.. ., IV(G)l), of the greatest difference (respectively the number of distinct differences) I~(v)-~(w)~ for 'VW E E(G)-We show that a graph on n vertices with bandsize k has bandwidth between k and cnl-f;, and that this is best possible. In the process we obtain best possible asymptotic bounds on the bandwidth of circulant graphs. The bandwidth and bandsize of random graphs are also compared, the former turning out to be n-cl logn and the latter at least ncz(logn)".

Lecture Notes in Computer Science, 2014

We introduce a concept of intersection dimension of a graph with respect to a graph class. This g... more We introduce a concept of intersection dimension of a graph with respect to a graph class. This generalizes Ferrers dimension, boxicity, and poset dimension, and leads to interesting new problems. We focus in particular on bipartite graph classes defined as intersection graphs of two kinds of geometric objects. We relate well-known graph classes such as interval bigraphs, two-directional orthogonal ray graphs, chain graphs, and (unit) grid intersection graphs with respect to these dimensions. As an application of these graphtheoretic results, we show that the recognition problems for certain graph classes are NP-complete.

TEMA - Tendências em Matemática Aplicada e Computacional, 2002

SIAM Journal on Discrete Mathematics, 2013

We describe a generating set for the variety of reflexive graphs that admit a compatible k-ary ne... more We describe a generating set for the variety of reflexive graphs that admit a compatible k-ary near-unanimity operation; we further delineate a very simple subset that generates the variety of j-absolute retracts; in particular we show that the class of reflexive graphs with a 4-NU operation coincides with the class of 3-absolute retracts. Our results generalise and encompass several results on NU-graphs and absolute retracts.

SIAM Journal on Computing, 1983

For arbitrary graphs G and H, a G-factor of H is a spanning subgraph of H composed of disjoint co... more For arbitrary graphs G and H, a G-factor of H is a spanning subgraph of H composed of disjoint copies of G. G-factors are natural generalizations of 1-factors (or perfect matchings), in which G replaces the complete graph on two vertices. Our results show that the perfect matching problem is essentially the only instance of the G-factor problem that is likely to admit a polynomial time bounded solution. Specifically, if G has any component with three or more vertices, then the existence question for G-factors is NP-complete. (In all other cases the question can be resolved in polynomial time.) The notion of a G-factor suggests a natural generalization where G is replaced by an arbitrary family of graphs. This generalization gives rise not only to further NP-completeness results but also to new polynomial algorithms and duality theorems extending results of the traditional theory of matching. An indication of the nature and scope of these new results is presented.

Journal of the London Mathematical Society, 1976

Journal of Graph Theory, 2003

Richard C. Brewster,1 Pavol Hell,2 Sarah H. Pantel,3 Romeo Rizzi,4 and Anders Yeo5 1DEPARTMENT OF... more Richard C. Brewster,1 Pavol Hell,2 Sarah H. Pantel,3 Romeo Rizzi,4 and Anders Yeo5 1DEPARTMENT OF COMPUTER SCIENCE BISHOP&amp;#x27;S UNIVERSITY, LENNOXVILLE QUÉ BEC, CANADA, JIM 1Z7 E-mail: rbrewste@ubishops.ca 2SCHOOL OF COMPUTING SCIENCE SIMON ...

Journal of Combinatorial Theory, Series B, 1986

The achromatic number of a graph G is the largest number of colors that can be assigned to the ve... more The achromatic number of a graph G is the largest number of colors that can be assigned to the vertices of G so that (i) adjacent vertices are assigned different colors, and (ii) any two different colors are assigned to some pair of adjacent vertices. We study the achromatic number from the point of view of computational complexity. We show that, for each fixed integer n, there is an algorithm which, for an arbitrary graph G, determines in time 0(1 E(G)1) whether the achromatic number of G is at least n. In contrast to this, we show that when n is part of the input, the problem is NP-complete even when restricted to bipartite graphs. The complexity of determining the achromatic number of a tree is unknown. We present polynomial algorithms to solve this problem for several classes of trees, and provide upper bounds on the achromatic number of a tree in terms of the maximum degree and the number of edges.