Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceeding

k-colored kernels

We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and (ii) for every $u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper, we prove that for every integer $k\geq 2$ there exists a $(k+1)$-colored digraph $D$ without $k$-colored kernel and if every directed cycle of an $m$-colored digraph is monochromatic, then it has a $k$-colored kernel for every positive integer $k.$ We obtain the following results for some generalizations of tournaments: (i) $m$-colored quasi-transitive and 3-quasi-transitive digraphs have a $k$% -colored kernel for every $k\geq 3$ and $k\geq 4,$ respectively (we conjecture that every $m$-colored $l$-quasi-transitive digraph has a $k$% -colored kernel for every $k\geq l+1)$, and (ii) $m$-colored locally in-tournament (out-tournament, respectively) digraphs have a $k$-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most $k$-colored

Nondeterministic graph property testing

A property of finite graphs is called nondeterministically testable if it has a "certificate" such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate, describe new application