30,244 research outputs found
Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs
for the class of interval graphs. We give a linear-time algorithm to find one
in any graph that is not an interval graph. Tucker characterized the minimal
forbidden submatrices of binary matrices that do not have the consecutive-ones
property. We give a linear-time algorithm to find one in any binary matrix that
does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th
International Workshop on Graph-Theoretic Concepts in Computer Scienc
Minimal Conflicting Sets for the Consecutive Ones Property in ancestral genome reconstruction
A binary matrix has the Consecutive Ones Property (C1P) if its columns can be
ordered in such a way that all 1's on each row are consecutive. A Minimal
Conflicting Set is a set of rows that does not have the C1P, but every proper
subset has the C1P. Such submatrices have been considered in comparative
genomics applications, but very little is known about their combinatorial
structure and efficient algorithms to compute them. We first describe an
algorithm that detects rows that belong to Minimal Conflicting Sets. This
algorithm has a polynomial time complexity when the number of 1's in each row
of the considered matrix is bounded by a constant. Next, we show that the
problem of computing all Minimal Conflicting Sets can be reduced to the joint
generation of all minimal true clauses and maximal false clauses for some
monotone boolean function. We use these methods on simulated data related to
ancestral genome reconstruction to show that computing Minimal Conflicting Set
is useful in discriminating between true positive and false positive ancestral
syntenies. We also study a dataset of yeast genomes and address the reliability
of an ancestral genome proposal of the Saccahromycetaceae yeasts.Comment: 20 pages, 3 figure
Subclasses of Normal Helly Circular-Arc Graphs
A Helly circular-arc model M = (C,A) is a circle C together with a Helly
family \A of arcs of C. If no arc is contained in any other, then M is a proper
Helly circular-arc model, if every arc has the same length, then M is a unit
Helly circular-arc model, and if there are no two arcs covering the circle,
then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit
Helly, normal Helly) circular-arc graph is the intersection graph of the arcs
of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model.
In this article we study these subclasses of Helly circular-arc graphs. We show
natural generalizations of several properties of (proper) interval graphs that
hold for some of these Helly circular-arc subclasses. Next, we describe
characterizations for the subclasses of Helly circular-arc graphs, including
forbidden induced subgraphs characterizations. These characterizations lead to
efficient algorithms for recognizing graphs within these classes. Finally, we
show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled
Proper Helly Circular-Arc Graphs) appeared at WG'0
Vertex adjacencies in the set covering polyhedron
We describe the adjacency of vertices of the (unbounded version of the) set
covering polyhedron, in a similar way to the description given by Chvatal for
the stable set polytope. We find a sufficient condition for adjacency, and
characterize it with similar conditions in the case where the underlying matrix
is row circular. We apply our findings to show a new infinite family of
minimally nonideal matrices.Comment: Minor revision, 22 pages, 3 figure
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