327 research outputs found

    Spanning spiders and light-splitting switches

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    AbstractMotivated by a problem in the design of optical networks, we ask when a graph has a spanning spider (subdivision of a star), or, more generally, a spanning tree with a bounded number of branch vertices. We investigate the existence of these spanning subgraphs in analogy to classical studies of Hamiltonicity

    Spanning trees with few branch vertices

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    A branch vertex in a tree is a vertex of degree at least three. We prove that, for all s≥1s\geq 1, every connected graph on nn vertices with minimum degree at least (1s+3+o(1))n(\frac{1}{s+3}+o(1))n contains a spanning tree having at most ss branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat

    On non-traceable, non-hypotraceable, arachnoid graphs

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    Motivated by questions concerning optical networks, in 2003 Gargano, Hammar, Hell, Stacho, and Vaccaro defined the notions of spanning spiders and arachnoid graphs. A spider is a tree with at most one branch (vertex of degree at least 3). The spider is centred at the branch vertex (if there is any,otherwise it is centred at any of the vertices). A graph is arachnoid if it has a spanning spider centred at any of its vertices. Traceable graphs are obviously arachnoid, and Gargano et al. observed that hypotraceable graphs (non-traceable graphs with the property that all vertex-deleted subgraphs are traceable) are also easily seen to be arachnoid. However, they did not find any other arachnoid graphs, and asked the question whether they exist. The main goal of this paper is to answer this question in the affirmative, moreover, we show that for any prescribed graph H, there exists a non-traceable, non-hypotraceable, arachnoid graph that contains H as an induced subgraph

    Spiders everywhere

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    A spider is a tree with at most one branch (a vertex of degree at least 3) centred at the branch if it exists, and centred at any vertex otherwise. A graph G is arachnoid if for any vertex v of G, there exists a spanning spider of G centred at v-in other words: there are spiders everywhere! Hypotraceable graphs are non-traceable graphs in which all vertex-deleted subgraphs are traceable. Gargano et al. (2004) defined arachnoid graphs as natural generalisations of traceable graphs and asked for the existence of arachnoid graphs that are (i) non-traceable and non-hypotraceable, or (ii) in which some vertex is the centre of only spiders with more than three legs. An affirmative answer to (ii) implies an affirmative answer to (i). While non-traceable, non-hypotraceable arachnoid graphs were described in Wiener (2017), (ii) remained open. In this paper we give an affirmative answer to this question and discuss spanning spiders whose legs must have some minimum length. (C) 2020 The Author(s). Published by Elsevier B.V

    Spanning k-trees and distance spectral radius in graphs

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    Let k≥2k\geq2 be an integer. A tree TT is called a kk-tree if dT(v)≤kd_T(v)\leq k for each v∈V(T)v\in V(T), that is, the maximum degree of a kk-tree is at most kk. Let λ1(D(G))\lambda_1(D(G)) denote the distance spectral radius in GG, where D(G)D(G) denotes the distance matrix of GG. In this paper, we verify a upper bound for λ1(D(G))\lambda_1(D(G)) in a connected graph GG to guarantee the existence of a spanning kk-tree in GG.Comment: 11 page

    An FPT Algorithm for Spanning Trees with Few Branch Vertices Parameterized by Modular-Width

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    The minimum branch vertices spanning tree problem consists in finding a spanning tree T of an input graph G having the minimum number of branch vertices, that is, vertices of degree at least three in T. This NP-hard problem has been widely studied in the literature and has many important applications in network design and optimization. Algorithmic and combinatorial aspects of the problem have been extensively studied and its fixed parameter tractability has been recently considered. In this paper we focus on modular-width and show that the problem of finding a spanning tree with the minimum number of branch vertices is FPT with respect to this parameter
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