A rooted-forest partition with uniform vertex demand

Katoh, Naoki; Tanigawa, Shin‐ichi

doi:10.1007/s10878-010-9367-x

Cited by 5 publications

References 25 publications

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A characterisation of the generic rigidity of 2-dimensional point–line frameworks

Jackson

Owen

2016

Journal of Combinatorial Theory, Series B

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A 2-dimensional point-line framework is a collection of points and lines in the plane which are linked by pairwise constraints that fix some angles between pairs of lines and also some point-line and pointpoint distances. It is rigid if every continuous motion of the points and lines which preserves the constraints results in a point-line framework which can be obtained from the initial framework by a translation or a rotation. We characterise when a generic point-line framework is rigid. Our characterisation gives rise to a polynomial algorithm for solving this decision problem.

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A characterisation of the generic rigidity of 2-dimensional point–line frameworks

Jackson

Owen

2016

Journal of Combinatorial Theory, Series B

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Antistrong digraphs

Bang‐Jensen

Bessy

Jackson

et al. 2017

Journal of Combinatorial Theory, Series B

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International audienceAn antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the arcs alternate between forward and backward arcs. An antidirected path is an antidirected trail where no vertex is repeated. We show that it is NP-complete to decide whether two vertices x, y in a digraph are connected by an antidirected path, while one can decide in linear time whether they are connected by an antidirected trail. A digraph D is antistrong if it contains an antidirected (x, y)-trail starting and ending with a forward arc for every choice of x, y ∈ V (D) . We show that antistrong connectivity can be decided in linear time. We discuss relations between antistrong connectivity and other properties of a digraph and show that the arc-minimal antistrong spanning subgraphs of a digraph are the bases of a matroid on its arc-set. We show that one can determine in polynomial time the minimum number of new arcs whose addition to D makes the resulting digraph the arc-disjoint union of k antistrong digraphs. In particular, we determine the minimum number of new arcs which need to be added to a digraph to make it antistrong. We use results from matroid theory to characterize graphs which have an antistrong orientation and give a polynomial time algorithm for constructing such an orientation when it exists. This immediately gives analogous results for graphs which have a connected bipartite 2-detachment. Finally, we study arc-decompositions of antistrong digraphs and pose several problems and conjectures

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Rooted-Tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries

Katoh¹,

Tanigawa²

2013

SIAM J. Discrete Math.

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As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose that we are given a graph G = (V, E), a multiset R = {r 1 , . . . , rt} of vertices in V , and a matroid M on R. We prove a necessary and sufficient condition for G to be decomposed into t edge-disjointIf M is a free matroid, this is a decomposition into t edge-disjoint spanning trees; thus, our result is a proper extension of NashWilliams' tree-partition theorem. Such a matroid constraint is motivated by combinatorial rigidity theory. As a direct application of our decomposition theorem, we present characterizations of the infinitesimal rigidity of frameworks with nongeneric "boundary," which extend classical the Laman's theorem for generic 2-rigidity of bar-joint frameworks and Tay's theorem for generic d-rigidity of body-bar frameworks. Introduction.In this paper two fundamental results in combinatorial optimization, the Tutte-Nash-Williams tree-packing theorem and the Nash-Williams tree-partition theorem, are extended. In 1961, Tutte [40] and Nash-Williams [26] independently proved that an undirected graph G = (V, E) contains k edge-disjoint spanning trees if and only if |δ G (P)| ≥ k|P| − k holds for any partition P of V , where δ G (P) denotes the set of edges of G connecting two distinct subsets of P and |P| denotes the number of subsets of P. As a dual form, the Nash-Williams tree-partition theorem [27] asserts that an undirected graph G = (V, E) can be decomposed into k edge-disjoint spanning trees if and only if |E| = k|V | − k and |F | ≤ k|V (F )| − k for any nonempty F ⊆ E, where V (F ) denotes the set of vertices incident to F . These two theorems are sometimes referred to in terms of rooted-edge-connectivity, as edge-disjoint spanning trees indicate how to send distinct "commodities" from a specific root-node to other vertices without interference. (In fact, the packing of spanning trees is a concept equivalent to the rooted-edge-connectivity orientation of undirected graphs; see, e.g., [9].) In this paper we address a more general situation. Suppose that we have t distinct roots, each of which has the ability to send a commodity, and suppose that the set of commodities possesses an independence structure, say, linear independence by regarding commodities as vectors. Then, we are asked

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A rooted-forest partition with uniform vertex demand

Cited by 5 publications

References 25 publications

A characterisation of the generic rigidity of 2-dimensional point–line frameworks

A characterisation of the generic rigidity of 2-dimensional point–line frameworks

Antistrong digraphs

Rooted-Tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries

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