99 research outputs found
Guarded Second-Order Logic, Spanning Trees, and Network Flows
According to a theorem of Courcelle monadic second-order logic and guarded
second-order logic (where one can also quantify over sets of edges) have the
same expressive power over the class of all countable -sparse hypergraphs.
In the first part of the present paper we extend this result to hypergraphs of
arbitrary cardinality. In the second part, we present a generalisation dealing
with methods to encode sets of vertices by single vertices
Monadic second-order definable graph orderings
We study the question of whether, for a given class of finite graphs, one can
define, for each graph of the class, a linear ordering in monadic second-order
logic, possibly with the help of monadic parameters. We consider two variants
of monadic second-order logic: one where we can only quantify over sets of
vertices and one where we can also quantify over sets of edges. For several
special cases, we present combinatorial characterisations of when such a linear
ordering is definable. In some cases, for instance for graph classes that omit
a fixed graph as a minor, the presented conditions are necessary and
sufficient; in other cases, they are only necessary. Other graph classes we
consider include complete bipartite graphs, split graphs, chordal graphs, and
cographs. We prove that orderability is decidable for the so called
HR-equational classes of graphs, which are described by equation systems and
generalize the context-free languages
Rank-decreasing transductions
We propose to study transformations on graphs, and more generally structures,
by looking at how the cut-rank (as introduced by Oum) of subsets is affected
when going from the input structure to the output structure. We consider
transformations in which the underlying sets are the same for both the input
and output, and so the cut-ranks of subsets can be easily compared. The purpose
of this paper is to give a characterisation of logically defined transductions
that is expressed in purely structural terms, without referring to logic:
transformations which decrease the cut-rank, in the asymptotic sense, are
exactly those that can be defined in monadic second-order logic. This
characterisation assumes that the transduction has inputs of bounded treewidth;
we also show that the characterisation fails in the absence of any assumptions.Comment: 22 pages, 13 figure
A coarse Tutte polynomial for hypermaps
We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction relations where the base case consists of isolated vertices. Our Tutte polynomial extends the classical Tutte polynomial of a graph as well as the Tutte polynomial of an embedded graph (i.e., the ribbon graph polynomial). We examine relations between our polynomial and other hypermap polynomials. We give hypermap duality and partial duality identities for our polynomial, as well as some evaluations
Algorithms for propositional model counting
AbstractWe present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worst-case time and space requirements
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