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In this paper, we consider the half-integral disjoint paths packing. For a graph G and k pairs of vertices (s1, t1), (s2, t2), …, (sk, tk) in G, the objective is to find paths P1, …, Pk in G such that Pi joins si and ti for i = 1, 2, …, k, and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with k = O((log n/log log n)1/7). This improves a result by Kleinberg [Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp 530-539] who proved the same conclusion when k = O((log log n)2/15). Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a "crossbar" of a polynomial size of the tree width of the graph. [ABSTRACT FROM AUTHOR]

We study the maximum edge-disjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2, … , sktk, the goal is to maximize the number of demands that can be connected (routed) by edge-disjoint paths. The natural multicommodity flow relaxation has an Ω(√n) integrality gap, where n is the number of nodes in G. Motivated by this, we consider solutions with small constant congestion c > 1, that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an O(log n) approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into well-linked subproblems. In this paper we obtain an O(1) approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura-Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation. [ABSTRACT FROM AUTHOR]

Two graphs G and H on n vertices are ϵ-far from being isomorphic if at least ϵ(n 2) edges must be added or removed from E(G) in order to make G and H isomorphic. In this paper we deal with the question of how many queries are required to distinguish between the case that two graphs are isomorphic and the case that they are ϵ-far from being isomorphic. A query is defined as probing the adjacency matrix of any one of the two graphs, i.e., asking if a pair of vertices forms an edge of the graph or not. We investigate both one-sided and two-sided error testers under two possible settings: The first setting is where both graphs need to be queried, and the second setting is where one of the graphs is fully known to the algorithm in advance. We prove that the query complexity of the best one-sided error testing algorithm is Θ(n3/2) if both graphs need to be queried, and that it is Θ(n) if one of the graphs is known in advance (where the Θ notation hides polylogarithmic factors in the upper bounds). For two-sided error testers, we prove that the query complexity of the best tester is Θ(√n) when one of the graphs is known in advance, and we show that the query complexity lies between Θ(n) and Õ(n5/4) if both G and H need to be queried. All of our algorithms are additionally nonadaptive, while all of our lower bounds apply for adaptive testers as well as nonadaptive ones. [ABSTRACT FROM AUTHOR]

The decycling number $\nabla(G)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resultant graph contains no cycles. In this paper, we study the decycling number for the family of graphs consisting of the Cartesian product of two cycles. We completely solve the problem of determining the decycling number of $C_m \square C_n$ for all $m$ and $n$. Moreover, we find a vertex set $T$ that yields a maximum induced tree in $C_m\square C_n$. [ABSTRACT FROM AUTHOR]

Recently, D. Corneil found a simple 3-sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil's algorithm to make it a certifying algorithm, and then describe a similar certifying 3-sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m+n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n). The certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs is [This symbol cannot be presented in ASCII format](√n), where n is the number of vertices in the graph (and [This symbol cannot be presented in ASCII format](f(n)) means Θ(f(n) · polylog(f(n)))). Thus there is a large gap between the complexity of testing in the two cases. In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity is [This symbol cannot be presented in ASCII format](min(√n, n²/m)), where m is the number of edges in the graph, and we match it with an almost tight lower bound. [ABSTRACT FROM AUTHOR]

Splitting off two edges su, sv in a graph G means deleting su, sv and adding a new edge uv. Let G = (V + s, E) be k-edge-connected in V (k ≥ 2) and let d(s) be even. Lovász proved that the edges incident to s can be split off in pairs in a such a way that the resulting graph on vertex set V is k-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to meet additional requirements. We prove structural properties of the set of those pairs u, v of neighbors of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type. By applying this method we obtain a short proof for a recent result of Nagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be k-edge-connected (l-edge-connected, respectively) in V (k, l ≥ 2) and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity in G (l-edge-connectivity in H, respectively), provided d(s) ≥ 6. If k and l are both even, then such a pair always exists. By using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size. [ABSTRACT FROM AUTHOR]

This paper gives a new and faster algorithm to find a 1-factor in a bipartite Δ-regular graph. The time complexity of this algorithm is O(nΔ + n log n log Δ), where n is the number of nodes. This implies an O(n log n log Δ + m log Δ) algorithm to edge-color a bipartite graph with n nodes, in edges, and maximum degree Δ. [ABSTRACT FROM AUTHOR]

Consider the 2n (or O(n²)) min-cut problems on a graph with n nodes and nonnegative edge weights. Gomory and Hu [J. Soc. Indust. Appl. Math., 9 (1961), pp. 551–570] showed (essentially) that there are at most n-1 different min-cuts. They also described a compact structure (the flow equivalent tree) of size O(n) with the following property: for any pair of nodes, the value of a min-cut can be obtained from this structure. Furthermore, they showed how this structure can be found by solving only n - 1 min-cut problems. This paper contains generalizations of these results. For example, consider a k-terminal cut problem on a graph: for a given set of k nodes, delete a minimum weight set of edges (called a k-cut) so that each of the k nodes is in a different component. There are kn (or O(nk)) such problems. Hassin [Math. Oper. Res., 13 (1988), pp. 535–542] showed (essentially) that there are at most k-1n-1 (or O(nk-1)) different min k-cuts. We describe a compact structure of size O(nk-1) with the following property: for any k nodes, the value of a min k-cut can be obtained from this structure. We also show how this structure can be found by solving only k-1n-1 k-terminal cut problems. This work builds upon the results of Hassin [Math. Oper. Res., 13 (1988), pp. 535–542], [Oper. Res. Lett., 9 (1990), pp. 315–318], and [Lecture Notes in Comput. Sci. 450, 1990, pp. 228–299]. [ABSTRACT FROM AUTHOR]

Bender, Richmond, and Wormald showed that in almost all planar 3-connected triangulations (or dually, 3-connected cubic maps) with n edges, the largest 4-connected triangulation (or dually, the largest cyclically 4-edge-connected cubic component) has about n/2 edges [Random Structures Algorithms, 7 (1995), pp. 273-285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps. [ABSTRACT FROM AUTHOR]

In the well-solved edge-connectivity augmentation problem we must find a minimum cardinality set F of edges to add to a given undirected graph to make it k-edge-connected. This paper solves the generalization where every edge of F must go between two different sets of a given partition of the vertex set. A special case of this partition-constrained problem, previously unsolved, is increasing the edge-connectivity of a bipartite graph to k while preserving bipartiteness. Based on this special case we present an application of our results in statics. Our solution to the general partition-constrained problem gives a min-max formula for F which includes as a special case the original min-max formula of Cai and Sun [Networks, 19 (1989), pp. 151-172] for the problem without partition constraints. When k is even the rain-max formula for the partition-constrained problem is a natural generalization of the unconstrained version. However, this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. We give a strongly polynomial algorithm that solves our problem in time O(n(m + n log n) log n). Here n and m denote the number of vertices and distinct edges of the given graph, respectively. This bound is identical to the best-known time bound for the problem without partition constraints. Our algorithm is based on the splitting off technique of Lovász, like several known efficient algorithms for the unconstrained problem. However, unlike previous splitting algorithms, when k is odd our algorithm must handle obstacles that prevent all edges from being split off. Our algorithm is of interest even when specialized to the unconstrained problem, because it produces an asymptotically optimum number of distinct splits. [ABSTRACT FROM AUTHOR]

In this paper we describe new constructions for de Bruijn sequences and Perfect Factors. These constructions are all based upon the idea of constructing one sequence (or set of sequences) from another. As a result of this fact, the sequences obtained from these construction methods possess simple decoding algorithms based on decoding the sequences used to construct them. Such decoding algorithms are of importance in position-location applications. [ABSTRACT FROM AUTHOR]