Inverse <Emphasis Type="Italic">p</Emphasis>-median problems with variable edge lengths

The inverse p-median problem with variable edge lengths on graphs is to modify the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified set of p vertices becomes a p-median with respect to the new edge lengths. The problem is shown to be strongly NP{\mathcal{NP}}-hard on general graphs and weakly NP{\mathcal{NP}}-hard on series-parallel graphs. Therefore, the special case on a tree is considered: It is shown that the inverse 2-median problem with variable edge lengths on trees is solvable in polynomial time. For the special case of a star graph we suggest a linear time algorithm.     

On the chordality of a graph

The chordality of a graph G = (V, E) is defined as the minimum k such that we can write E = E₁ ∩ … ∩ E_k with each (V, E_i) a chordal graph. We present several results bounding the value of this generalization of boxicity. Our principal result is that the chordality of a graph is at most its tree width. In particular, series-parallel graphs have chordality at most 2. Potential strengthenings of this statement fail in that there are planar graphs with chordality 3 and series-parallel graphs with boxicity 3. © 1993 John Wiley & Sons, Inc.     

A branch-and-cut algorithm based on semidefinite programming for the minimum <Emphasis Type="Italic">k</Emphasis>-partition problem

The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results; and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing a fast exact approach for k≥3.     

Approximation algorithms for finding and partitioning unit-disk graphs into co-<Emphasis Type="Italic">k</Emphasis>-plexes

This article studies a degree-bounded generalization of independent sets called co-k-plexes. Constant factor approximation algorithms are developed for the maximum co-k-plex problem on unit-disk graphs. The related problem of minimum co-k-plex coloring that generalizes classical vertex coloring is also studied in the context of unit-disk graphs. We extend several classical approximation results for independent sets in UDGs to co-k-plexes, and settle a recent conjecture on the approximability of co-k-plex coloring in UDGs.     

Aggregation approach for the minimum binary cost tension problem

The aggregation technique, dedicated to two-terminal series–parallel graphs (TTSP-graphs) and introduced lately to solve the minimum piecewise linear cost tension problem, is adapted here to solve the minimum binary cost tension problem (BCT problem). Even on TTSP-graphs, the BCT problem has been proved to be NP-complete. As far as we know, the aggregation is the only algorithm, with mixed integer programming (MIP), proposed to solve exactly the BCT problem on TTSP-graphs. A comparison of the efficiency of both methods and a heuristic is presented.     

Local and variable neighborhood search for the <Emphasis Type="Italic">k</Emphasis>-cardinality subgraph problem

The minimum weighted k-cardinality subgraph problem consists of finding a connected subgraph of a given graph with exactly k edges whose sum of weights is minimum. For this NP-hard combinatorial problem, only constructive types of heuristics have been suggested in the literature. In this paper we propose a new heuristic based on variable neighborhood search metaheuristic rules. This procedure uses a new local search developed by us. Extensive numerical results that include graphs with up to 5,000 vertices are reported. It appears that VNS outperforms all previous methods.     

A common generalization of line graphs and clique graphs

Both the line graph and the clique graph are defined as intersection graphs of certain families of complete subgraphs of a graph. We generalize this concept. By a k-edge of a graph we mean a complete subgraph with k vertices or a clique with fewer than k vertices. The k-edge graph Δ_k(G) of a graph G is defined as the intersection graph of the set of all k-edges of G. The following three problems are investigated for k-edge graphs. The first is the characterization problem. Second, sets of graphs closed under the k-edge graph operator are found. The third problem is the question of convergence: What happens to a graph if we take iterated k-edge graphs?     

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree Δ. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang–Swendsen–Kotecký (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k11Δ/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3k<Δ/(20logΔ), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when kΔ/2, provided k is larger than some absolute constant k₀.     

Register Allocation in Structured Programs

In this article we look at the register allocation problem. In the literature this problem is frequently reduced to the general graph coloring problem and the solutions to the problem are obtained from graph coloring heuristics. Hence, no algorithm with a good performance guarantee is known. Here we show that when attention is restricted tostructured programswhich we define to be programs whose control-flow graphs are series-parallel, there is an efficient algorithm that produces a solution which is within a factor of 2 of the optimal solution. We note that even with the previous restriction the resulting coloring problem is NP-complete.We also consider how to delete a minimum number of edges from arbitrary control-flow graphs to make them series-parallel and to apply our algorithm. We show that this problem is Max SNP hard. However, we define the notion of anapproximate articulation pointand we give efficient algorithms to find approximate articulation points. We present a heuristic for the edge deletion problem based on this notion which seems to work well when the given graph is close to series-parallel.     

Planarization and fragmentability of some classes of graphs

The coefficient of fragmentability of a class of graphs measures the proportion of vertices that need to be removed from the graphs in the class in order to leave behind bounded sized components. We have previously given bounds on this parameter for the class of graphs satisfying a given constant bound on maximum degree. In this paper, we give fragmentability bounds for some classes of graphs of bounded average degree, as well as classes of given thickness, the class of k-colourable graphs, and the class of n-dimensional cubes. In order to establish the fragmentability results for bounded average degree, we prove that the proportion of vertices that must be removed from a graph of average degree at most in order to leave behind a planar subgraph (in fact, a series-parallel subgraph) is at most , provided or the graph is connected and . The proof yields an algorithm for finding large induced planar subgraphs and (under certain conditions) a lower bound on the size of the induced planar subgraph it finds. This bound is similar in form to the one we found for a previous algorithm we developed for that problem, but applies to a larger class of graphs.     

Stronger linear programming relaxations of max-cut

We consider linear programming relaxations for the max cut problem in graphs, based on k-gonal inequalities. We show that the integrality ratio for random dense graphs is asymptotically 1+1/k and for random sparse graphs is at least 1+3/k. There are O(n^k)k-gonal inequalities. These results generalize work by Poljak and Tuza, who gave similar results for k=3.     

PARTITION A GRAPH WITH SMALL DIAMETER INTO TWO INDUCED MATCHINGS

§1　IntroductionGraphs considered in this paper are finite and simple.For a graph G,its vertex setandedge set are denoted by V(G) and E(G) ,respectively.If vertices u and v are connected inG,the distance between u and v,denoted by d G(u,v) ,is the length ofa shortest(u,v) -pathin G.The diameter of a connected graph G is the maximum distance between two verticesof G.For X V(G) ,the neighbor set NG(X) of X is defined byNG(X) ={ y∈V(G) \X:there is x∈X such thatxy∈E(G) } .NG({ x} )…     

On stability of Hamilton‐connectedness under the 2‐closure in claw‐free graphs

It is easy to characterize chordal graphs by every k‐cycle having at least f(k) = k ? 3 chords. I prove new, analogous characterizations of the house‐hole‐domino‐free graphs using f(k) = 2?(k ? 3)/2?, and of the graphs whose blocks are trivially perfect using f(k) = 2k ? 7. These three functions f(k) are optimum in that each class contains graphs in which every k‐cycle has exactly f(k) chords. The functions 3?(k ? 3)/3? and 3k ? 11 also characterize related graph classes, but without being optimum. I consider several other graph classes and their optimum functions, and what happens when k‐cycles are replaced with k‐paths. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:137‐147, 2011     

Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an “almost uniform” partition is called an (l,u)-partition. We deal with three problems to find an (l,u)-partition of a given graph; the minimum partition problem is to find an (l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l,u)-partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O(u⁴n) and the p-partition problem can be solved in time O(p²u⁴n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.     

Det‐extremal cubic bipartite graphs

Let G be a connected k–regular bipartite graph with bipartition V(G) = X ∪ Y and adjacency matrix A. We say G is det‐extremal if per (A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det‐extremal 3‐connected cubic bipartite graphs. We extend McCuaig's result by determining the structure of det‐extremal cubic bipartite graphs of connectivity two. We use our results to determine which numbers can occur as orders of det‐extremal connected cubic bipartite graphs, thus solving a problem due to H. Gropp. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 50–64, 2003     

Complexity of clique‐coloring odd‐hole‐free graphs

In this paper we investigate the problem of clique‐coloring, which consists in coloring the vertices of a graph in such a way that no monochromatic maximal clique appears, and we focus on odd‐hole‐free graphs. On the one hand we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, but on the other hand it is NP‐hard to decide if they are 2‐clique‐colorable, and we do not know if there exists any bound k₀ such that they are all k₀ ‐clique‐colorable. First we will prove that (odd hole, codiamond)‐free graphs are 2‐clique‐colorable. Then we will demonstrate that the complexity of 2‐clique‐coloring odd‐hole‐free graphs is actually Σ₂ P‐complete. Finally we will study the complexity of deciding whether or not a graph and all its subgraphs are 2‐clique‐colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 139–156, 2009     

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

An Algorithm to Construct <Emphasis Type="BoldItalic">k</Emphasis>-Regular <Emphasis Type="BoldItalic">k</Emphasis>-Connected Graphs with the Maximum <Emphasis Type="BoldItalic">k</Emphasis>-Diameter

 Let f(2m,k) be the Maximum k-diameter of k-regular k-connected graphs on 2m vertices. In this paper we give an algorithm and prove that we can construct k-regular k-connected graphs on 2m vertices with the maximum k-diameter using it. We also prove some known results about f(2m,k) and verify that we can get some unknown values of f(2m,k) by our algorithm. Received: December 1, 2000 Final version received: March 12, 2002 Acknowledgments. We thank the referee for many useful suggestions.     

A Solution to a Problem of Jacobson,Kézdy and Lehel

We solve a problem proposed by Jacobson, Kézdy, and Lehel [4] concerning the existence of forbidden induced subgraph characterizations of line graphs of linear k-uniform hypergraphs with sufficiently large minimal edge-degree. Actually, we prove that for each k3 there is a finite set Z(k) of graphs such that each graph G with minimum edge-degree at least 2k²–3k+1 is the line graph of a linear k-uniform hypergraph if and only if G is a Z(k)-free graph.Acknowledgments. We thank the anonymous referees, whose suggestions helped to improve the presentation of the paper.Winter 2002/2003 DIMACS Award is gratefully acknowledged2000 Mathematics Subject Classification: 05C65 (05C75, 05C85)     

On <Emphasis Type="Italic">k</Emphasis>-pairable graphs from trees

The concept of the k-pairable graphs was introduced by Zhibo Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter p(G), called the pair length of a graph G, as the maximum k such that G is k-pairable and p(G) = 0 if G is not k-pairable for any positive integer k. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees G with p(G) = 1 and prove that p(G □ H) = p(G) + p(H) when both G and H are trees.