A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.

. For any fixed rational parameter t 1, a (tree) t--spanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General t--spanners and their variants have multiple applications in the field of communication networks, distributed systems, and network design. In this paper, we combine the two concepts of simple structured, sparse t--spanners and faulttolerance by examining independent tree t--spanners. Given a root vertex r, this is a pair of tree t--spanners, such that the two paths from any vertex to r are edge disjoint or internally vertex dijoint, respectively. It is shown that a pair of independent tree t--spanners can be found in linear time for t ! 3, whereas the problem for arbitrary t 4 is NP--complete. As a less restrictive concept, we also treat tree t--root-spanners, where the distance constraint is relaxed. Here, we show that the problem of finding an independent pair of su...

We prove that given any fixed edge ra in a 4-connected graph G, there exists a cycle C through ra such that G − (V (C) − {r}) is 2-connected. This will provide the first step in a decomposition for 4-connected graphs. We also prove that for any given edge e in a 5-connected graph G there exists an induced cycle C through e in G such that G − V (C) is 2-connected. This provides evidence for a conjecture of Lovász.

F20.22> 6;6 and K 6;7 are 3-choosable (and simplified the proof that K 5;8 is 3-choosable), thereby completing the proof that n(3) = 14. They also proved n(k) k \Delta n(k \Gamma 2) + 2 k , which improves the previously known upper bounds (from m(k)) when k is even. Zolt'an Furedi proved that K 5;13 is not 3-choosable, so the remaining complete bipartite graphs for which 3-choosability has not been settled are K 5;t for 9 t 12 and K 6;t for 8 t 10. [6] also observes that K k;t is k-choosable if and only if t ! k<F

. We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths. Keywords. independent spanning trees, line digraphs, vertex-connectivity, de Bruijn digraphs, Kautz digraphs, interconnection networks, broadcasting. 1 Introduction Unless stated otherwise each digraph of this paper is nite and may have loops but not multiarcs. Let G be a digraph. Then V (G) and A(G) denote the vertex set and the arc set of G, respectively. Let (u; v) 2 A(G). Then we say that u is adjacent to v, and v is adjacent from u. Also, it is said that (u; v) is incident to v and incident from u. Let (v; w) 2 A(G). Then we say that (u; v) is adjacent to (v; w), and (v; w) is adjacent from (u; v). Let ...

In this paper, we describe an O(|V (G) | 2) algorithm for finding a “non-separating planar chain” in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into “planar chains”. This work was motivated by the study of a multi-tree approach to reliability in distributed networks, as well as the study of non-separating induced paths in highly connected graphs.

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An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have λ arc-disjoint arborescences rooted at r. A similar theorem of Menger guarantees λ strongly arc disjoint rv-paths for every vertex v, where “strongly ” means no two paths contain a pair of symmetric arcs. We prove that if a directed graph D contains two arc-disjoint spanning arborescences rooted at r, then D contains two such arborences with the property that for every node v the paths from r to v in the two arborences satisfy Menger‘s theorem.

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