7,299 research outputs found
Induced minors and well-quasi-ordering
A graph is an induced minor of a graph if it can be obtained from an
induced subgraph of by contracting edges. Otherwise, is said to be
-induced minor-free. Robin Thomas showed that -induced minor-free
graphs are well-quasi-ordered by induced minors [Graphs without and
well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 --
247, 1985].
We provide a dichotomy theorem for -induced minor-free graphs and show
that the class of -induced minor-free graphs is well-quasi-ordered by the
induced minor relation if and only if is an induced minor of the gem (the
path on 4 vertices plus a dominating vertex) or of the graph obtained by adding
a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved
two decomposition theorems which are of independent interest.
Similar dichotomy results were previously given for subgraphs by Guoli Ding
in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502,
1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and
well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]
Forbidden Directed Minors and Kelly-width
Partial 1-trees are undirected graphs of treewidth at most one. Similarly,
partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known
that an undirected graph is a partial 1-tree if and only if it has no K_3
minor. In this paper, we generalize this characterization to partial 1-DAGs. We
show that partial 1-DAGs are characterized by three forbidden directed minors,
K_3, N_4 and M_5
A General Framework for Well-Structured Graph Transformation Systems
Graph transformation systems (GTSs) can be seen as wellstructured transition
systems (WSTSs), thus obtaining decidability results for certain classes of
GTSs. In earlier work it was shown that wellstructuredness can be obtained
using the minor ordering as a well-quasiorder. In this paper we extend this
idea to obtain a general framework in which several types of GTSs can be seen
as (restricted) WSTSs. We instantiate this framework with the subgraph ordering
and the induced subgraph ordering and apply it to analyse a simple access
rights management system.Comment: Extended version (including proofs) of a paper accepted at CONCUR
201
Dual Feynman transform for modular operads
We introduce and study the notion of a dual Feynman transform of a modular
operad. This generalizes and gives a conceptual explanation of Kontsevich's
dual construction producing graph cohomology classes from a contractible
differential graded Frobenius algebra. The dual Feynman transform of a modular
operad is indeed linear dual to the Feynman transform introduced by Getzler and
Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman
transform, the dual notion admits an extremely simple presentation via
generators and relations; this leads to an explicit and easy description of its
algebras. We discuss a further generalization of the dual Feynman transform
whose algebras are not necessarily contractible. This naturally gives rise to a
two-colored graph complex analogous to the Boardman-Vogt topological tree
complex.Comment: 27 pages. A few conceptual changes in the last section; in particular
we prove that the two-colored graph complex is a resolution of the
corresponding modular operad. It is now called 'BV-resolution' as suggested
by Sasha Vorono
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