32,932 research outputs found
Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2
Deciding whether a given graph has a square root is a classical problem that
has been studied extensively both from graph theoretic and from algorithmic
perspectives. The problem is NP-complete in general, and consequently
substantial effort has been dedicated to deciding whether a given graph has a
square root that belongs to a particular graph class. There are both
polynomial-time solvable and NP-complete cases, depending on the graph class.
We contribute with new results in this direction. Given an arbitrary input
graph G, we give polynomial-time algorithms to decide whether G has an
outerplanar square root, and whether G has a square root that is of pathwidth
at most 2
Grad and Classes with Bounded Expansion II. Algorithmic Aspects
Classes of graphs with bounded expansion are a generalization of both proper
minor closed classes and degree bounded classes. Such classes are based on a
new invariant, the greatest reduced average density (grad) of G with rank r,
∇r(G). These classes are also characterized by the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. These results lead to several new linear time algorithms, such as an
algorithm for counting all the isomorphs of a fixed graph in an input graph or
an algorithm for checking whether there exists a subset of vertices of a priori
bounded size such that the subgraph induced by this subset satisfies some
arbirtrary but fixed first order sentence. We also show that for fixed p,
computing the distances between two vertices up to distance p may be performed
in constant time per query after a linear time preprocessing. We also show,
extending several earlier results, that a class of graphs has sublinear
separators if it has sub-exponential expansion. This result result is best
possible in general
Tilings of quadriculated annuli
Tilings of a quadriculated annulus A are counted according to volume (in the
formal variable q) and flux (in p). We consider algebraic properties of the
resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p
must be roots of unity and for q > 0, real negative.Comment: 33 pages, 12 figures; Minor changes were made to make some passages
cleare
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