33,073 research outputs found
Advice Complexity of the Online Induced Subgraph Problem
Several well-studied graph problems aim to select a largest (or smallest)
induced subgraph with a given property of the input graph. Examples of such
problems include maximum independent set, maximum planar graph, and many
others. We consider these problems, where the vertices are presented online.
With each vertex, the online algorithm must decide whether to include it into
the constructed subgraph, based only on the subgraph induced by the vertices
presented so far. We study the properties that are common to all these problems
by investigating the generalized problem: for a hereditary property \pty, find
some maximal induced subgraph having \pty. We study this problem from the point
of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we
give a tight trade-off relationship stating that for inputs of length n roughly
n/c bits of advice are both needed and sufficient to obtain a solution with
competitive ratio c, regardless of the choice of \pty, for any c (possibly a
function of n). Surprisingly, a similar result cannot be obtained for the
symmetric problem: for a given cohereditary property \pty, find a minimum
subgraph having \pty. We show that the advice complexity of this problem varies
significantly with the choice of \pty.
We also consider preemptive online model, where the decision of the algorithm
is not completely irreversible. In particular, the algorithm may discard some
vertices previously assigned to the constructed set, but discarded vertices
cannot be reinserted into the set again. We show that, for the maximum induced
subgraph problem, preemption cannot help much, giving a lower bound of
bits of advice needed to obtain competitive ratio ,
where is any increasing function bounded by \sqrt{n/log n}. We also give a
linear lower bound for c close to 1
A note on 2--bisections of claw--free cubic graphs
A \emph{--bisection} of a bridgeless cubic graph is a --colouring
of its vertex set such that the colour classes have the same cardinality and
all connected components in the two subgraphs induced by the colour classes
have order at most . Ban and Linial conjectured that {\em every bridgeless
cubic graph admits a --bisection except for the Petersen graph}.
In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
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