247 research outputs found
Simultaneous Embeddability of Two Partitions
We study the simultaneous embeddability of a pair of partitions of the same
underlying set into disjoint blocks. Each element of the set is mapped to a
point in the plane and each block of either of the two partitions is mapped to
a region that contains exactly those points that belong to the elements in the
block and that is bounded by a simple closed curve. We establish three main
classes of simultaneous embeddability (weak, strong, and full embeddability)
that differ by increasingly strict well-formedness conditions on how different
block regions are allowed to intersect. We show that these simultaneous
embeddability classes are closely related to different planarity concepts of
hypergraphs. For each embeddability class we give a full characterization. We
show that (i) every pair of partitions has a weak simultaneous embedding, (ii)
it is NP-complete to decide the existence of a strong simultaneous embedding,
and (iii) the existence of a full simultaneous embedding can be tested in
linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201
Logical limit laws for minor-closed classes of graphs
Let be an addable, minor-closed class of graphs. We prove that
the zero-one law holds in monadic second-order logic (MSO) for the random graph
drawn uniformly at random from all {\em connected} graphs in on
vertices, and the convergence law in MSO holds if we draw uniformly at
random from all graphs in on vertices. We also prove analogues
of these results for the class of graphs embeddable on a fixed surface,
provided we restrict attention to first order logic (FO). Moreover, the
limiting probability that a given FO sentence is satisfied is independent of
the surface . We also prove that the closure of the set of limiting
probabilities is always the finite union of at least two disjoint intervals,
and that it is the same for FO and MSO. For the classes of forests and planar
graphs we are able to determine the closure of the set of limiting
probabilities precisely. For planar graphs it consists of exactly 108
intervals, each of length . Finally, we analyse
examples of non-addable classes where the behaviour is quite different. For
instance, the zero-one law does not hold for the random caterpillar on
vertices, even in FO.Comment: minor changes; accepted for publication by JCT
Gaps in the Saturation Spectrum of Trees
A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from the complement of G to G results in a copy of H. The minimum number of edges (the size) of an H-saturated graph on n vertices is denoted sat(n, H), while the maximum size is the well studied extremal number, ex(n, H). The saturation spectrum for a graph H is the set of sizes of H-saturated graphs between sat(n, H) and ex(n, H). In this paper we show that paths, trees with a vertex adjacent to many leaves, and brooms have a gap in the saturation spectrum
Digraph Coloring Games and Game-Perfectness
In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix
Tree-Verifiable Graph Grammars
Hyperedge-Replacement grammars (HR) have been introduced by Courcelle in
order to extend the notion of context-free sets from words and trees to graphs
of bounded tree-width. While for words and trees the syntactic restrictions
that guarantee that the associated languages of words resp. trees are regular -
and hence, MSO-definable - are known, the situation is far more complicated for
graphs. Here, Courcelle proposed the notion of regular graph grammars, a
syntactic restriction of HR grammars that guarantees the definability of the
associated languages of graphs in Counting Monadic Second Order Logic (CMSO).
However, these grammars are not complete in the sense that not every
CMSO-definable set of graphs of bounded tree-width can be generated by a
regular graph grammar. In this paper, we introduce a new syntactic restriction
of HR grammars, called tree-verifiable graph grammars, and a new notion of
bounded tree-width, called embeddable bounded tree-width, where the later
restricts the trees of a tree-decomposition to be a subgraph of the analyzed
graph. The main property of tree-verifiable graph grammars is that their
associated languages are CMSO-definable and that the have bounded embeddable
tree-width. We show further that they strictly generalize the regular graph
grammars of Courcelle. Finally, we establish a completeness result, showing
that every language of graphs that is CMSO-definable and of bounded embeddable
tree-width can be generated by a tree-verifiable graph grammar
On graph classes with minor-universal elements
A graph is universal for a graph class , if every is a minor of . We prove the existence or absence of universal
graphs in several natural graph classes, including graphs component-wise
embeddable into a surface, and graphs forbidding , or , or
as a minor. We prove the existence of uncountably many minor-closed
classes of countable graphs that (do and) do not have a universal element.
Some of our results and questions may be of interest to the finite graph
theorist. In particular, one of our side-results is that every -minor-free
graph is a minor of a -minor-free graph of maximum degree 22
Subgraph densities in a surface
Given a fixed graph that embeds in a surface , what is the
maximum number of copies of in an -vertex graph that embeds in
? We show that the answer is , where is a
graph invariant called the `flap-number' of , which is independent of
. This simultaneously answers two open problems posed by Eppstein
(1993). When is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem
fully rewritten, fixes a serious error in the previous version found by Kevin
Hendre
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