Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a greedoid if and only if G has a unique perfect matching.Comment: 12 pages, 12 figure

On the Core of a Unicyclic Graph

A set S is independent in a graph G if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. A connected graph having only one cycle, say C, is a unicyclic graph. In this paper we prove that if G is a unicyclic graph of order n and n-1 = alpha(G) + mu(G), then core(G) coincides with the union of cores of all trees in G-C.Comment: 8 pages, 5 figure

Abelian networks IV. Dynamics of nonhalting networks

An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. This paper is a continuation of the abelian networks series of Bond and Levine (2016), for which we extend the theory of abelian networks that halt on all inputs to networks that can run forever. A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the \emph{torsion group} of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as $$\sum_{n \geq 1} r_n z^n = \det (\frac{1}{1-z}D - A )$$ where $r_n$ is the number of recurrent chip-and-rotor configurations with $n$ chips; $D$ is the diagonal matrix of outdegrees, and $A$ is the adjacency matrix. A consequence is that the sequence $(r_n)_{n \geq 1}$ completely determines the spectrum of the simple random walk on the network.Comment: 95 pages, 21 figure