A randomized embedding algorithm for trees

In this paper, we propose a simple and natural randomized algorithm to embed a tree T in a given graph G. The algorithm can be viewed as a “self-avoiding tree-indexed random walk“. The order of the tree T can be as large as a constant fraction of the order of the graph G, and the maximum degree of T can be close to the minimum degree of G. We show that our algorithm works in a variety of interesting settings. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of order εd ² and maximum degree at most d-2εd-2. As there exist d-regular graphs without 4-cycles and with O(d ²) vertices, this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth and graphs with no complete bipartite subgraph K _s,t .     

Families of cuts with the MFMC-property

A family ℱ of cuts of an undirected graphG=(V, E) is known to have the weak MFMC-property if (i) ℱ is the set ofT-cuts for someT⊆V with |T| even, or (ii) ℱ is the set of two-commodity cuts ofG, i.e. cuts separating any two distinguished pairs of vertices ofG, or (iii) ℱ is the set of cuts induced (in a sense) by a ring of subsets of a setT⊆V. In the present work we consider a large class of families of cuts of complete graphs and prove that a family from this class has the MFMC-property if and only if it is one of (i), (ii), (iii).     

Two characterizations of chain partitioned probe graphs

Chain graphs are exactly bipartite graphs without induced 2K ₂ (a graph with four vertices and two disjoint edges). A graph G=(V,E) with a given independent set S⊆V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: G is a chain partitioned probe graph if and only if the enhanced graph G ^* is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m ²)-time recognition algorithm for chain partitioned probe graphs.     

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ _G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.     

Strong chromatic index of subset graphs

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S_m(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.     

Maximal Energy Bipartite Graphs

 Given a graph G, its energy E(G) is defined to be the sum of the absolute values of the eigenvalues of G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. Here we show that if G is a bipartite graph with n vertices, then

On the Nullity of Bipartite Graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. We obtain some lower bounds for the nullity of graphs and we then find the nullity of bipartite graphs with no cycle of length a multiple of 4 as a subgraph. Among bipartite graphs on n vertices, the star has the greatest nullity (equal to n − 2). We generalize this by showing that among bipartite graphs with n vertices, e edges and maximum degree Δ which do not have any cycle of length a multiple of 4 as a subgraph, the greatest nullity is . G. R. Omidi: This research was in part supported by a grant from IPM (No.87050016).     

An algorithm for finding hamilton paths and cycles in random graphs

This paper describes a polynomial time algorithm HAM that searches for Hamilton cycles in undirected graphs. On a random graph its asymptotic probability of success is that of the existence of such a cycle. If all graphs withn vertices are considered equally likely, then using dynamic programming on failure leads to an algorithm with polynomial expected time. The algorithm HAM is also used to solve the symmetric bottleneck travelling salesman problem with probability tending to 1, asn tends to ∞. Various modifications of HAM are shown to solve several Hamilton path problems. Supported by NSF Grant MCS 810 4854.     

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, we obtain a polynomial delay algorithm for listing the anti‐vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems, including d‐factors in bipartite graphs, and perfect 2‐matchings in general graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 209–232, 2006     

Minimum Color Sum of Bipartite Graphs

The problem ofminimum color sumof a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated withinn^{1 − ε}, for any ε > 0, unlessNP = ZPP(Bar-Noyet al., Information and Computation140(1998), 183–202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unlessP = NP. The proof is byL-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.     

The struction of a graph: Application toCN-free graphs

We consider the class of graphs characterized by the forbidden subgraphsC andN:C is the claw (unique graph with degree sequence (3, 1, 1, 1)) andN the net (unique graph with degree sequence (3, 3, 3, 1, 1, 1)). For this class of graphs (calledCN-free) an algorithm is described for determining the stability numberα(G). It is based on a construction associating with anyCN-free graphG anotherCN-free graphG′ such thatα(G′)=α(G)−1. Such a construction reducing the stability number is called a struction. This work was completed while this author was visiting the Dept. of Combinatories and Optimization at the University of Waterloo, Ontario.     

Bipartite Divisor Graphs for Integer Subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.     

On the harmonious coloring of collections of graphs

The hermonious coloring number of the graph G, HC(G), is the smallest number of colors needed to label the vertices of G such that adjacent vertices received different colors and no two edges are incident with the same color pair. In this paper, we investigate the HC-number of collections of disjoint paths, cycles, complete graphs, and complete bipartite graphs. We determine exact expressions for the HC-number of collections of paths and 4m-cycles. © 1995, John Wiley & Sons, Inc.     

Complete partitions of graphs

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [19], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/√lgn). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C·cp(G)/√lgn classes, then NP ⊆ RTime(n ^{O(lg lg n)}). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn) ^c ) for some constant c strictly between 0 and 1. The work reported here is a merger of the results reported in [30] and [21].     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

A characterization of chain probe graphs

A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+F is a chain graph (i.e., a 2K ₂-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n ²) recognition algorithm.     

Quick Approximation to Matrices and Applications

m ×n matrix A with entries between say −1 and 1, and an error parameter ε between 0 and 1, we find a matrix D (implicitly) which is the sum of simple rank 1 matrices so that the sum of entries of any submatrix (among the ) of (A−D) is at most εmn in absolute value. Our algorithm takes time dependent only on ε and the allowed probability of failure (not on m, n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, R?dl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. From our matrix approximation, the above graph algorithms and the Regularity Lemma and several other results follow in a simple way. We generalize our approximations to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems. Received: July 25, 1997     

Expansion Of Product Replacement Graphs

We establish a connection between the expansion coefficient of the product replacement graph Γ_k(G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs Γ_k(PSL(2,p)) form an expander family, under assumption that all Cayley graphs of PSL(2,p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [42,52].     

On the Convexity Number of Graphs

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.