28,132 research outputs found
Max flow vitality in general and -planar graphs
The \emph{vitality} of an arc/node of a graph with respect to the maximum
flow between two fixed nodes and is defined as the reduction of the
maximum flow caused by the removal of that arc/node. In this paper we address
the issue of determining the vitality of arcs and/or nodes for the maximum flow
problem. We show how to compute the vitality of all arcs in a general
undirected graph by solving only max flow instances and, In
-planar graphs (directed or undirected) we show how to compute the vitality
of all arcs and all nodes in worst-case time. Moreover, after
determining the vitality of arcs and/or nodes, and given a planar embedding of
the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in
time proportional to the size of the set.Comment: 12 pages, 3 figure
Topological transition in disordered planar matching: combinatorial arcs expansion
In this paper, we investigate analytically the properties of the disordered
Bernoulli model of planar matching. This model is characterized by a
topological phase transition, yielding complete planar matching solutions only
above a critical density threshold. We develop a combinatorial procedure of
arcs expansion that explicitly takes into account the contribution of short
arcs, and allows to obtain an accurate analytical estimation of the critical
value by reducing the global constrained problem to a set of local ones. As an
application to a toy representation of the RNA secondary structures, we suggest
generalized models that incorporate a one-to-one correspondence between the
contact matrix and the RNA-type sequence, thus giving sense to the notion of
effective non-integer alphabets.Comment: 28 pages, 6 figures, published versio
Planar and Poly-Arc Lombardi Drawings
In Lombardi drawings of graphs, edges are represented as circular arcs, and
the edges incident on vertices have perfect angular resolution. However, not
every graph has a Lombardi drawing, and not every planar graph has a planar
Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be
drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi
drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing
and further investigate topics connecting planarity and Lombardi drawings.Comment: Expanded version of paper appearing in the 19th International
Symposium on Graph Drawing (GD 2011). 16 pages, 8 figure
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
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