7,409 research outputs found
Finding an induced subdivision of a digraph
We consider the following problem for oriented graphs and digraphs: Given an
oriented graph (digraph) , does it contain an induced subdivision of a
prescribed digraph ? The complexity of this problem depends on and on
whether must be an oriented graph or is allowed to contain 2-cycles. We
give a number of examples of polynomial instances as well as several
NP-completeness proofs
Restricted frame graphs and a conjecture of Scott
Scott proved in 1997 that for any tree , every graph with bounded clique
number which does not contain any subdivision of as an induced subgraph has
bounded chromatic number. Scott also conjectured that the same should hold if
is replaced by any graph . Pawlik et al. recently constructed a family
of triangle-free intersection graphs of segments in the plane with unbounded
chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This
shows that Scott's conjecture is false whenever is obtained from a
non-planar graph by subdividing every edge at least once.
It remains interesting to decide which graphs satisfy Scott's conjecture
and which do not. In this paper, we study the construction of Pawlik et al. in
more details to extract more counterexamples to Scott's conjecture. For
example, we show that Scott's conjecture is false for any graph obtained from
by subdividing every edge at least once. We also prove that if is a
2-connected multigraph with no vertex contained in every cycle of , then any
graph obtained from by subdividing every edge at least twice is a
counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our
results to an appendix
Permanents, Pfaffian orientations, and even directed circuits
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in
such a way that the permanent of A equals the determinant of the modified
matrix? When does a real square matrix have the property that every real matrix
with the same sign pattern (that is, the corresponding entries either have the
same sign or are both zero) is nonsingular? When is a hypergraph with n
vertices and n hyperedges minimally nonbipartite? When does a bipartite graph
have a "Pfaffian orientation"? Given a digraph, does it have no directed
circuit of even length? Given a digraph, does it have a subdivision with no
even directed circuit?
It is known that all of the above problems are equivalent. We prove a
structural characterization of the feasible instances, which implies a
polynomial-time algorithm to solve all of the above problems. The structural
characterization says, roughly speaking, that a bipartite graph has a Pfaffian
orientation if and only if it can be obtained by piecing together (in a
specified way) planar bipartite graphs and one sporadic nonplanar bipartite
graph.Comment: 47 pages, published versio
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