110,799 research outputs found
Note on Perfect Forests
A spanning subgraph of a graph is called perfect if is a forest,
the degree of each vertex in is odd, and each tree of is
an induced subgraph of . We provide a short proof of the following theorem
of A.D. Scott (Graphs & Combin., 2001): a connected graph contains a
perfect forest if and only if has an even number of vertices
Note on Perfect Forests in Digraphs
A spanning subgraph of a graph is called {\em perfect} if is a
forest, the degree of each vertex in is odd, and each tree of
is an induced subgraph of . Alex Scott (Graphs \& Combin., 2001) proved
that every connected graph contains a perfect forest if and only if has
an even number of vertices. We consider four generalizations to directed graphs
of the concept of a perfect forest. While the problem of existence of the most
straightforward one is NP-hard, for the three others this problem is
polynomial-time solvable. Moreover, every digraph with only one strong
component contains a directed forest of each of these three generalization
types. One of our results extends Scott's theorem to digraphs in a non-trivial
way
Characterizing perfect recall using next-step temporal operators in S5 and sub-S5 Epistemic Temporal Logic
We review the notion of perfect recall in the literature on interpreted
systems, game theory, and epistemic logic. In the context of Epistemic Temporal
Logic (ETL), we give a (to our knowledge) novel frame condition for perfect
recall, which is local and can straightforwardly be translated to a defining
formula in a language that only has next-step temporal operators. This frame
condition also gives rise to a complete axiomatization for S5 ETL frames with
perfect recall. We then consider how to extend and consolidate the notion of
perfect recall in sub-S5 settings, where the various notions discussed are no
longer equivalent
Fine-grained dichotomies for the Tutte plane and Boolean #CSP
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of
evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard
almost everywhere, and the remaining points admit polynomial-time algorithms.
Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination
with Curticapean [7], extended the #P-hardness results to tight lower bounds
under the counting exponential time hypothesis #ETH, with the exception of the
line , which was left open. We complete the dichotomy theorem for the
Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs
of a given -vertex graph cannot be determined in time unless
#ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann
[6] for counting the number of satisfying assignments to a constraint
satisfaction problem instance over the Boolean domain. We prove that all
#P-hard cases are also hard under #ETH. The main ingredient is to prove that
the number of independent sets in bipartite graphs with vertices cannot be
computed in time unless #ETH fails. In order to prove our results,
we use the block interpolation idea by Curticapean [7] and transfer it to
systems of linear equations that might not directly correspond to
interpolation.Comment: 16 pages, 1 figur
Burning a Graph is Hard
Graph burning is a model for the spread of social contagion. The burning
number is a graph parameter associated with graph burning that measures the
speed of the spread of contagion in a graph; the lower the burning number, the
faster the contagion spreads. We prove that the corresponding graph decision
problem is \textbf{NP}-complete when restricted to acyclic graphs with maximum
degree three, spider graphs and path-forests. We provide polynomial time
algorithms for finding the burning number of spider graphs and path-forests if
the number of arms and components, respectively, are fixed.Comment: 20 Pages, 4 figures, presented at GRASTA-MAC 2015 (October 19-23rd,
2015, Montr\'eal, Canada
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