310 research outputs found
Degree-doubling graph families
Let G be a family of n-vertex graphs of uniform degree 2 with the property
that the union of any two member graphs has degree four. We determine the
leading term in the asymptotics of the largest cardinality of such a family.
Several analogous problems are discussed.Comment: 9 page
Eigenvalues and forbidden subgraphs I
We present sharp inequalities relating the number of vertices, edges, and
triangles of a graph to the smallest eigenvalue of its adjacency matrix and the
largest eigenvalue of its Laplacian.Comment: Some calculation errors in the first version are correcte
Seymour's second neighborhood conjecture for tournaments missing a generalized star
Seymour's Second Neighborhood Conjecture asserts that every digraph (without
digons) has a vertex whose first out-neighborhood is at most as large as its
second out-neighborhood. We prove its weighted version for tournaments missing
a generalized star. As a consequence the weighted version holds for tournaments
missing a sun, star, or a complete graph.Comment: Accepted for publication in Journal of Graph Theory in 24 June 201
On Backtracking in Real-time Heuristic Search
Real-time heuristic search algorithms are suitable for situated agents that
need to make their decisions in constant time. Since the original work by Korf
nearly two decades ago, numerous extensions have been suggested. One of the
most intriguing extensions is the idea of backtracking wherein the agent
decides to return to a previously visited state as opposed to moving forward
greedily. This idea has been empirically shown to have a significant impact on
various performance measures. The studies have been carried out in particular
empirical testbeds with specific real-time search algorithms that use
backtracking. Consequently, the extent to which the trends observed are
characteristic of backtracking in general is unclear. In this paper, we present
the first entirely theoretical study of backtracking in real-time heuristic
search. In particular, we present upper bounds on the solution cost exponential
and linear in a parameter regulating the amount of backtracking. The results
hold for a wide class of real-time heuristic search algorithms that includes
many existing algorithms as a small subclass
Graph Abstraction and Abstract Graph Transformation
Many important systems like concurrent heap-manipulating programs, communication networks, or distributed algorithms are hard to verify due to their inherent dynamics and unboundedness. Graphs are an intuitive representation of states of these systems, where transitions can be conveniently described by graph transformation rules.
We present a framework for the abstraction of graphs supporting abstract graph transformation. The abstraction method naturally generalises previous approaches to abstract graph transformation. The set of possible abstract graphs is finite. This has the pleasant consequence of generating a finite transition system for any start graph and any finite set of transformation rules. Moreover, abstraction preserves a simple logic for expressing properties on graph nodes. The precision of the abstraction can be adjusted according to properties expressed in this logic to be verified
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