1,131 research outputs found

    A note on forbidding clique immersions

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    Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without KtK_t-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without K⃗t\vec{K}_t-immersions, where K⃗t\vec{K}_t denotes the bidirected complete digraph of order tt

    Nondeterministic graph property testing

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    A property of finite graphs is called nondeterministically testable if it has a "certificate" such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate, describe new application

    Strong Connectivity in Directed Graphs under Failures, with Application

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    In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let GG be a digraph with mm edges and nn vertices, and let G∖eG\setminus e be the digraph obtained after deleting edge ee from GG. As a first result, we show how to compute in O(m+n)O(m+n) worst-case time: (i)(i) The total number of strongly connected components in G∖eG\setminus e, for all edges ee in GG. (ii)(ii) The size of the largest and of the smallest strongly connected components in G∖eG\setminus e, for all edges ee in GG. Let GG be strongly connected. We say that edge ee separates two vertices xx and yy, if xx and yy are no longer strongly connected in G∖eG\setminus e. As a second set of results, we show how to build in O(m+n)O(m+n) time O(n)O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: (i)(i) Report in O(n)O(n) worst-case time all the strongly connected components of G∖eG\setminus e, for a query edge ee. (ii)(ii) Test whether an edge separates two query vertices in O(1)O(1) worst-case time. (iii)(iii) Report all edges that separate two query vertices in optimal worst-case time, i.e., in time O(k)O(k), where kk is the number of separating edges. (For k=0k=0, the time is O(1)O(1)). All of the above results extend to vertex failures. All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the 11-connectivity (i.e., 11-edge and 11-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of GG with O(n)O(n) edges that maintains the 11-connectivity cuts of GG and the decompositions induced by those cuts.Comment: An extended abstract of this work appeared in the SODA 201

    Single Source - All Sinks Max Flows in Planar Digraphs

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    Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

    Sink-Stable Sets of Digraphs

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    We introduce the notion of sink-stable sets of a digraph and prove a min-max formula for the maximum cardinality of the union of k sink-stable sets. The results imply a recent min-max theorem of Abeledo and Atkinson on the Clar number of bipartite plane graphs and a sharpening of Minty's coloring theorem. We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on cyclic stable sets
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