Abstract. The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on counting functions of generalized colorings. We show that this class encompasses the examples of graph polynomials from the literature. Furthermore, we extend the definition of graph polynomials definable in MSOL to allow definability in full second order, SOL. Finally, we show that the SOL-definable graph polynomials extended with a combinatorial counting function are exactly the counting functions of generalized colorings definable in SOL.
The domination polynomial D(G, x) of a graph G is the generating function of its dominating sets. We prove that D(G, x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G, x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G, x) based on articulation vertices, and more generally, on splitting sets of vertices.where G i are obtained from G using various vertex and edge elimination operations and the g i (x)'s are given rational functions. For example, it is well-known that the independence polynomial satisfies a linear recurrence relation with respect to two vertex elimination operations, the deletion of a vertex and the deletion of vertex's closed neighborhood. Other prominent graph polynomials in the literature satisfy similar recurrence relations with respect to vertex and edge elimination operations, among them the matching polynomial, the chromatic polynomial and the vertex-cover polynomial, see e.g. [11].In contrast, it is significantly harder to find recurrence relations for the domination polynomial. We show in Theorem 2.4 that D(G, x) does not satisfy any linear recurrence relation which applies only the commonly used vertex operations of deletion, extraction, contraction and neighborhood-contraction. Nor does D(G, x) satisfy any linear recurrence relation using only edge deletion, contraction and extraction.In spite of this non-existence result, we give in this paper an abundance of recurrence relations and splitting formulas for the domination polynomial.The domination polynomial was studied recently by several authors, see [1,2,3,5,6,7,8,9,10,12]. The previous research focused mainly on the roots of domination polynomials and on the domination polynomials of various classes of special graphs. In [12] it is shown that computing the domination polynomial D(G, x) of a graph G is NPhard and some examples for graphs for which D(G, x) can be computed efficiently are given. Some of our results, e.g. Theorem 5.14, lead to efficient schemes to compute the domination polynomial.An outline of the paper is as follows. In Section 2 we give a recurrence relation for arbitrary graphs. In Section 3 we give simple recurrence steps in special cases, which allow us e.g. to dispose of triangles, induced 5-paths and irrelevant edges. In Section 4 we consider graphs of connectivity 1, and give several splitting formulas for them. In Section 5 we generalize the results of the previous section to arbitrary separating vertex sets. In Section 6 we show a recurrence relation for arbitrary graphs which uses derivatives of domination polynomials.
We study the complexity of the model-checking problem for parameterized discrete-timed systems with arbitrarily many anonymous and identical contributors, with and without a distinguished "controller" process. Processes communicate via synchronous rendezvous. Our work extends the seminal work on untimed systems [German, Sistla: Reasoning about Systems with Many Processes. J. ACM 39(3), 1992] by the addition of discrete-time clocks, thus allowing one to model more realistic protocols.For the case without a controller, we show that the systems can be efficiently simulated -and vice versa -by systems of untimed processes that communicate via rendezvous and symmetric broadcast, which we call "RB-systems". Symmetric broadcast is a novel communication primitive that, like ordinary asymmetric broadcast allows all processes to synchronize; however, it has no distinction between sender/receiver processes.We show that the complexity of the parameterized model-checking problem for safety specifications is pspace-complete, and for liveness specifications it is decidable and in exptime. The latter result is proved using automata theory, rational linear programming, and geometric reasoning for solving certain reachability questions in a new variant of vector addition systems called "vector rendezvous systems". We believe these proof techniques are of independent interest and will be useful in solving related problems.For the case with a controller, we show that the parameterized model-checking problems for RB-systems and systems with asymmetric broadcast as a primitive are inter-reducible. This allows us to prove that for discrete timed-networks with a controller the parameterized model-checking problem is undecidable for liveness specifications.Our work exploits the intimate and fruitful connection between parameterized discretetimed systems and systems of processes communicating via broadcast. This allows us to provide a rare and surprising decidability result for liveness properties of parameterized timed-systems, as well as extend work from untimed systems to timed systems.
Abstract.A graph polynomial p(G,X) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific pointsX =x0. In this paper we study the question how to prove that a given graph parameter, say ω(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte polynomial, the interlace polynomial, or any graph polynomial of some infinite family of graph polynomials.Our result is very general. We give a sufficient condition in terms of the connection matrix of graph parameter f (G) which implies that it cannot be the evaluation of any graph polynomial which is invariantly definable in CMSOL, the Monadic Second Order Logic augmented with modular counting quantifiers. This criterion covers most of the graph polynomials known from the literature.
Parameterized model checking is the problem of deciding if a given formula holds irrespective of the number of participating processes. A standard approach for solving the parameterized model checking problem is to reduce it to model checking finitely many finite-state systems. This work considers the theoretical power and limitations of this technique. We focus on concurrent systems in which processes communicate via pairwise rendezvous, as well as the special cases of disjunctive guards and token passing; specifications are expressed in indexed temporal logic without the next operator; and the underlying network topologies are generated by suitable formulas and graph operations. First, we settle the exact computational complexity of the parameterized model checking problem for some of our concurrent systems, and establish new decidability results for others. Second, we consider the cases where model checking the parameterized system can be reduced to model checking some fixed number of processes, the number is known as a cutoff. We provide many cases for when such cutoffs can be computed, establish lower bounds on the size of such cutoffs, and identify cases where no cutoff exists. Third, we consider cases for which the parameterized system is equivalent to a single finite-state system (more precisely a Büchi word automaton), and establish tight bounds on the sizes of such automata.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.