Constructing the Spectrum of Packings and Coverings for the Complete Graph with Stars with up to Five Edges

The packing and covering problems have been considered for several classes of graphs. For instance, Bryant et. al. have investigated the packing problem for paths and cycles, and the packing and covering problems for 3-cubes. The packing and covering problems were settled for stars with up to six edges by Roditty. In this paper, for every possible leave graph (excess graph), we find a corresponding maximum packing (minimum covering) of the complete graph with stars with up to five edges.     

Equitable partitions into matchings and coverings in mixed graphs

Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings.In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching–covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the problems of partitioning a mixed graph into matching forests or mixed edge covers, so that all parts are equal with respect to some criterion, such as edge/arc numbers or total sizes. Moreover, we provide the best possible multicriteria equalization.     

The relation of time indexed formulations of single machine scheduling problems to the node packing problem

 The relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known results on the facial structure of the scheduling problem. Previous work on the facial structure has focused on describing the convex hull of the set of feasible partial schedules, schedules in which not all jobs have to be started. The equivalence between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is determined by the parameters of the scheduling problem is established. The main contribution of this paper is to show that the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5-hole inequalities of the convex hull of the set of feasible node packings in this graph respectively. Received: September 10, 2000 / Accepted: April 20, 2002 Published online: September 27, 2002 Key words. scheduling – node packing – polyhedral methods – facet defining graphs – lifted valid inequalities – facet inducing inequalities}     

Error graphs and the reconstruction of elements in groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions.In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology [P.A. Pevzner, Computational Molecular Biology: An Algorithmic Approach, MIT Press, Cambridge, MA, 2000; D. Sankoff, N. El-Mabrouk, Genome rearrangement, in: T. Jiang, T. Smith, Y. Xu, M.Q. Zhang (Eds.), Current Topics in Computational Molecular Biology, MIT Press, 2002]. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.     

On minimal forbidden subgraph characterizations of balanced graphs

A {0, 1}-matrix is balanced if it contains no square submatrix of odd order with exactly two 1's per row and per column. Balanced matrices lead to ideal formulations for both set packing and set covering problems. Balanced graphs are those graphs whose clique-vertex incidence matrix is balanced.While a forbidden induced subgraph characterization of balanced graphs is known, there is no such characterization by minimal forbidden induced subgraphs. In this work we provide minimal forbidden induced subgraph characterizations of balanced graphs restricted to some graph classes which also lead to polynomial time or even linear time recognition algorithms within the corresponding subclasses.     

Feasible insertions in job shop scheduling,short cycles and stable sets

Insertion problems arise in scheduling when additional activities have to be inserted into a given schedule. This paper investigates insertion problems in a general disjunctive scheduling framework capturing a variety of job shop scheduling problems and insertion types. First, a class of scheduling problems is introduced, characterized by disjunctive graphs with the so-called short cycle property, and it is shown that in such problems, the feasible selections correspond to the stable sets of maximum cardinality in an associated conflict graph. Two types of insertion problems are then identified where the underlying disjunctive graph is through- or bi-connected. For these cases, it is shown that the short cycle property holds and the conflict graph is bipartite, allowing to derive a polyhedral characterization of all feasible insertions. An efficient method for deciding whether there exists a feasible insertion, and a lower and upper bound procedure for the minimum makespan insertion problem are developed. For bi-connected graphs, this procedure solves the insertion problem to optimality. The obtained results are applied to three extensions of the classical Job Shop, the Multi-Processor Task, Blocking and No-Wait Job Shop, and two types of insertions, job and block insertion.     

On external-memory MST, SSSP and multi-way planar graph separation

Recently external memory graph problems have received considerable attention because massive graphs arise naturally in many applications involving massive data sets. Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open.The results in this paper fall in two main classes. First we develop an improved algorithm for the problem of computing a minimum spanning tree (MST) of a general undirected graph. Second we show that on planar undirected graphs the problems of computing a multi-way graph separation and single source shortest paths (SSSP) can be reduced I/O-efficiently to planar breadth-first search (BFS). Since BFS can be trivially reduced to SSSP by assigning all edges weight one, it follows that in external memory planar BFS, SSSP, and multi-way separation are equivalent. That is, if any of these problems can be solved I/O-efficiently, then all of them can be solved I/O-efficiently in the same bound. Our planar graph results have subsequently been used to obtain I/O-efficient algorithms for all fundamental problems on planar undirected graphs.     

Grid Graphs,Gorenstein Polytopes,and Domino Stackings

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics.     

Complexity of the packing coloring problem for trees

Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i-th class have pairwise distance greater than i. The main result of this paper is a solution of an open problem of Goddard et al. showing that the decision whether a tree allows a packing coloring with at most k classes is NP-complete.We further discuss specific cases when this problem allows an efficient algorithm. Namely, we show that it is decideable in polynomial time for graphs of bounded treewidth and diameter, and fixed parameter tractable for chordal graphs.We accompany these results by several observations on a closely related variant of the packing coloring problem, where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers.     

More on quasi-random graphs,subgraph counts and graph limits

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases.The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.     

Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a naive linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal–dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem. These results match the currently known best results for purely edge-weighted graphs.     

Cuts, matrix completions and graph rigidity

This paper brings together several topics arising in distinct areas: polyhedral combinatorics, in particular, cut and metric polyhedra; matrix theory and semidefinite programming, in particular, completion problems for positive semidefinite matrices and Euclidean distance matrices; distance geometry and structural topology, in particular, graph realization and rigidity problems. Cuts and metrics provide the unifying theme. Indeed, cuts can be encoded as positive semidefinite matrices (this fact underlies the approximative algorithm for max-cut of Goemans and Williamson) and both positive semidefinite and Euclidean distance matrices yield points of the cut polytope or cone, after applying the functions 1/π arccos(.) or √. When fixing the dimension in the Euclidean distance matrix completion problem, we find the graph realization problem and the related question of unicity of realization, which leads to the question of graph rigidity. Our main objective here is to present in a unified setting a number of results and questions concerning matrix completion, graph realization and rigidity problems. These problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross-fertilization between the various fields involved.     

On covering graphs by complete bipartite subgraphs

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m²/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.     

Homomorphism–homogeneous graphs

We answer two open questions posed by Cameron and Nesetril concerning homomorphism–homogeneous graphs. In particular we show, by giving a characterization of these graphs, that extendability to monomorphism or to homomorphism leads to the same class of graphs when defining homomorphism–homogeneity. Further, we show that there are homomorphism–homogeneous graphs that do not contain the Rado graph as a spanning subgraph answering the second open question. We also treat the case of homomorphism–homogeneous graphs with loops allowed, showing that the corresponding decision problem is co–NP complete. Finally, we extend the list of considered morphism–types and show that the graphs for which monomorphisms can be extended to epimor‐phisms are complements of homomorphism–homogeneous graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 253–261, 2010     

Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with a constant capacity for all plants. These facet inequalities depend on the capacity and thus differ fundamentally from the valid inequalities for the uncapacited version of the problem.We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.This research was partially supported by Grant # ECS-8316224 from the National Science Foundation's Program in Systems Theory and Operations Research.     

List Homomorphisms and Circular Arc Graphs

G , H, and lists , a list homomorphism of G to H with respect to the lists L is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Received: July 22, 1996/Revised: Revised June 10, 1998     

Arc transitive covering digraphs and their eigenvalues

We prove that every finite regular digraph has an arc-transitive covering digraph (whose arcs are equivalent under automorphisms) and every finite regular graph has a 2-arc-transitive covering graph. As a corollary, we sharpen C. D. Godsil's results on eigenvalues and minimum polynomials of vertex-transitive graphs and digraphs. Using Godsil's results, we prove, that given an integral matrix A there exists an arc-transitive digraph X such that the minimum polynomial of A divides that of X. It follows that there exist arc-transitive digraphs with nondiagonalizable adjacency matrices, answering a problem by P. J. Cameron. For symmetric matrices A, we construct a 2-arc-transitive graphs X.     

Chordal bipartite completion of colored graphs

Golumbic, Kaplan, and Shamir [Graph sandwich problems, J. Algorithms 19 (1995) 449-473], in their paper on graph sandwich problems published in 1995, left the status of the sandwich problems for strongly chordal graphs and chordal bipartite graphs open. It was recently shown [C.M.H. de Figueiredo, L. Faria, S. Klein, R. Sritharan, On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs, Theoret. Comput. Sci., accepted for publication] that the sandwich problem for strongly chordal graphs is NP-complete. We show that given graph G with a proper vertex coloring c, determining whether there is a supergraph of G that is chordal bipartite and also is properly colored by c is NP-complete. This implies that the sandwich problem for chordal bipartite graphs is also NP-complete.     

Every triangle-free planar graph has a planar upward drawing

A planar ordered set has a triangle-free, planar covering graph; on the other hand, there are nonplanar ordered sets whose covering graphs are planar. We show thatevery triangle-free planar graph has a planar upward drawing. This planar upward drawing can be constructed in time, polynomial in the number of vertices.Our results shed light on the apparently difficult problem, of long-standing, whether there is aneffective planarity-testing procedure for an ordered set.Supported in part by the Alexander von Humboldt Stiftung.Supported in part by the Deutsche Forschungsgemeinschaft.