327 research outputs found
Spanning spiders and light-splitting switches
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
A branch vertex in a tree is a vertex of degree at least three. We prove
that, for all , every connected graph on vertices with minimum
degree at least contains a spanning tree having at most
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
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
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
Let be an integer. A tree is called a -tree if
for each , that is, the maximum degree of a -tree is at most .
Let denote the distance spectral radius in , where
denotes the distance matrix of . In this paper, we verify a upper bound for
in a connected graph to guarantee the existence of a
spanning -tree in .Comment: 11 page
An FPT Algorithm for Spanning Trees with Few Branch Vertices Parameterized by Modular-Width
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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