141 research outputs found
Keeping Avoider's graph almost acyclic
We consider biased Avoider-Enforcer games in the monotone and strict
versions. In particular, we show that Avoider can keep his graph being a forest
for every but maybe the last round of the game if . By this
we obtain essentially optimal upper bounds on the threshold biases for the
non-planarity game, the non--colorability game, and the -minor game
thus addressing a question and improving the results of Hefetz, Krivelevich,
Stojakovi\'c, and Szab\'o. Moreover, we give a slight improvement for the lower
bound in the non-planarity game.Comment: 11 page
Fast winning strategies in Avoider-Enforcer games
In numerous positional games the identity of the winner is easily determined.
In this case one of the more interesting questions is not {\em who} wins but
rather {\em how fast} can one win. These type of problems were studied earlier
for Maker-Breaker games; here we initiate their study for unbiased
Avoider-Enforcer games played on the edge set of the complete graph on
vertices. For several games that are known to be an Enforcer's win, we
estimate quite precisely the minimum number of moves Enforcer has to play in
order to win. We consider the non-planarity game, the connectivity game and the
non-bipartite game
Are there any good digraph width measures?
Several different measures for digraph width have appeared in the last few
years. However, none of them shares all the "nice" properties of treewidth:
First, being \emph{algorithmically useful} i.e. admitting polynomial-time
algorithms for all \MS1-definable problems on digraphs of bounded width. And,
second, having nice \emph{structural properties} i.e. being monotone under
taking subdigraphs and some form of arc contractions. As for the former,
(undirected) \MS1 seems to be the least common denominator of all reasonably
expressive logical languages on digraphs that can speak about the edge/arc
relation on the vertex set.The latter property is a necessary condition for a
width measure to be characterizable by some version of the cops-and-robber game
characterizing the ordinary treewidth. Our main result is that \emph{any
reasonable} algorithmically useful and structurally nice digraph measure cannot
be substantially different from the treewidth of the underlying undirected
graph. Moreover, we introduce \emph{directed topological minors} and argue that
they are the weakest useful notion of minors for digraphs
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
First-Order Logic with Connectivity Operators
First-order logic (FO) can express many algorithmic problems on graphs, such
as the independent set and dominating set problem, parameterized by solution
size. On the other hand, FO cannot express the very simple algorithmic question
of whether two vertices are connected. We enrich FO with connectivity
predicates that are tailored to express algorithmic graph properties that are
commonly studied in parameterized algorithmics. By adding the atomic predicates
that hold true in a graph if there exists a
path between (the valuations of) and after (the valuations of)
have been deleted, we obtain separator logic .
We show that separator logic can express many interesting problems such as
the feedback vertex set problem and elimination distance problems to
first-order definable classes. We then study the limitations of separator logic
and prove that it cannot express planarity, and, in particular, not the
disjoint paths problem. We obtain the stronger disjoint-paths logic
by adding the atomic predicates that evaluate to true if there are internally vertex disjoint paths
between (the valuations of) and for all .
Disjoint-paths logic can express the disjoint paths problem, the problem of
(topological) minor containment, the problem of hitting (topological) minors,
and many more. Finally, we compare the expressive power of the new logics with
that of transitive closure logics and monadic second-order logic.Comment: 18 pages, 3 figure
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