1,652 research outputs found
Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration
We continue research into a well-studied family of problems that ask whether
the vertices of a graph can be partitioned into sets and~, where is
an independent set and induces a graph from some specified graph class
. We let be the class of -degenerate graphs. This
problem is known to be polynomial-time solvable if (bipartite graphs) and
NP-complete if (near-bipartite graphs) even for graphs of maximum degree
. Yang and Yuan [DM, 2006] showed that the case is polynomial-time
solvable for graphs of maximum degree . This also follows from a result of
Catlin and Lai [DM, 1995]. We consider graphs of maximum degree on
vertices. We show how to find and in time for , and in
time for . Together, these results provide an algorithmic
version of a result of Catlin [JCTB, 1979] and also provide an algorithmic
version of a generalization of Brook's Theorem, which was proven in a more
general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007].
Moreover, the two results enable us to complete the complexity classification
of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex
colouring reconfiguration graph between two given -colourings of a graph
of maximum degree
Partitioning a triangle-free planar graph into a forest and a forest of bounded degree
An -partition of a graph is a vertex-partition into
two sets and such that the graph induced by is a forest and the
one induced by is a forest with maximum degree at most . We prove that
every triangle-free planar graph admits an -partition.
Moreover we show that if for some integer there exists a triangle-free
planar graph that does not admit an -partition, then it
is an NP-complete problem to decide whether a triangle-free planar graph admits
such a partition.Comment: 16 pages, 12 figure
Decomposing a triangle-free planar graph into a forest and a subcubic forest
We strengthen a result of Dross, Montassier and Pinlou (2017) that the vertex
set of every triangle-free planar graph can be decomposed into a set that
induces a forest and a set that induces a forest with maximum degree at most
, showing that can be replaced by .Comment: 8 pages. This version corrects an error in the previous version
(where we made a false claim at the end of the proof of Theorem 2), without
changing the overall structure of the proo
Computing the Girth of a Planar Graph in Linear Time
The girth of a graph is the minimum weight of all simple cycles of the graph.
We study the problem of determining the girth of an n-node unweighted
undirected planar graph. The first non-trivial algorithm for the problem, given
by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and
Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further
reduced the running time to O(n log n). In this paper, we solve the problem in
O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin
Generic singular configurations of linkages
We study the topological and differentiable singularities of the
configuration space C(\Gamma) of a mechanical linkage \Gamma in d-dimensional
Euclidean space, defining an inductive sufficient condition to determine when a
configuration is singular. We show that this condition holds for generic
singularities, provide a mechanical interpretation, and give an example of a
type of mechanism for which this criterion identifies all singularities
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