First-order logic axiomatization of metric graph theory

European Journal of Combinatorics, 2009

A fundamental notion in metric graph theory is that of the interval function I : V × V → 2 V − {∅} of a (finite) connected graph G = (V, E), where I(u, v) = { w | d(u, w) + d(w, v) = d(u, v) } is the interval between u and v. An obvious question is whether I can be characterized in a nice way amongst all functions F : V × V → 2 V − {∅}. This was done in [13, 14, 16] by axioms in terms of properties of the functions F. The authors of the present paper, in the conviction that characterizing the interval function belongs to the central questions of metric graph theory, return here to this result again. In this characterization the set of axioms consists of five simple, and obviously necessary, axioms, already presented in [9], plus two more complicated axioms. The question arises whether the last two axioms are really necessary in the form given or whether simpler axioms would do the trick. This question turns out to be non-trivial. The aim of this paper is to show that these two supplementary axioms are optimal in the following sense. The functions satisfying only the five simple axioms are studied extensively. Then the obstructions are pinpointed why such functions may not be the interval function of some connected graph. It turns out that these obstructions occur precisely when either one of the supplementary axioms is not satisfied. It is also shown that each of these supplementary axioms is independent of the other six axioms. The presented way of proving the characterizing theorem (Theorem 3 here) allows us to find two new separate "intermediate" results (Theorems 1 and 2). In addition some new characterizations of modular and median graphs are presented. As shown in the last section the results of this paper could provide a new perspective on finite connected graphs.

Discrete Mathematics, 1978

In particular, if S is a finite set then M is ca3led a (finite) matroid axiom (4) is redundant.

For a fixed set X, an arbitrary weight structure d ∈ [0, ∞] X×X can be inter-preted as a distance assignment between pairs of points on X. Restrictions (i.e., metric axioms) on the behaviour of any such d naturally arise, such as separation, triangle inequality and symmetry. We present an order-theoretic investigation of various collections of weight structures, as naturally occurring subsets of [0, ∞] X×X satisfying certain metric axioms. Furthermore, we exploit the categorical notion of adjunctions when investigating connections between the above collections of weight structures. As a corollary, we present several lattice-embeddability theorems on a well-known collection of weight structures on X.

Information and Computation, 2009

In this paper, we study the (positive) graph relational calculus. The basis for this calculus was introduced by Curtis and Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easy to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

European Journal of Combinatorics, 2007

We bring together algebraic concepts such as equational class and various concepts from graph theory for developing a structure theory for graphs that promotes a deeper analysis of their metric properties. The basic operators are Cartesian multiplication and gated amalgamation or, alternatively, retraction. Specifically, finite weakly median graphs are known to be decomposable (relative to these operators) into smaller pieces that in turn are parts of hyperoctahedra, the pentagonal pyramid, or of certain triangulations of the plane. This decomposition scheme can be interpreted as Birkhoff's subdirect representation in purely algebraic terms.

2011

We identify the locally finite graphs that are quantifier-eliminable and their first order theories in the signature of distance predicates.

During the last few years, my research has been in graph theory and has led to several publications and pre-publications . Firstly I will introduce the results of and .

ArXiv, 2020

In this paper we consider certain types of betweenness axioms on the interval function $I_G$ of a connected graph $G$. We characterize the class of graphs for which $I_G$ satisfy these axioms. The class of graphs that we characterize include the important class of Ptolemaic graphs and some proper superclasses of Ptolemaic graphs: the distance hereditary graphs and the bridged graphs. We also provide axiomatic characterizations of the interval function of these classes of graphs using an arbitrary function known as \emph{transit function}.

Discrete Mathematics, 1999

A corrected proof is given for the existence of a universal countable {Cj, CS,. , Czii ,)-free graph. We also prove that there is a universal countable w-free graph. There is no universal countable H-free graph if H is the disjoint union of 3 or more complete n-cliques for some n 22, plus one vertex, joined to every other point.

We present the basic ideas of forms (a generalization of Ehresmann's sketches) and their theories and models, more explicitly than in previous expositions. Forms provide the ability to specify mathematical structures and data types in any appropriate category, including many types of structures (e.g. function spaces) that cannot be specified by sketches. We also outline a new kind of formal logic (based on graphs instead of strings of symbols) that gives an intrinsically categorial definition of assertion and proof for each type of form. This formal logic is new to this monograph. The relationship between multisorted equational logic and finite product theories is worked out in detail.

A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.

Discrete Mathematics, 1992

PrCa, P., Graphs and topologies on discrete sets, Discrete Mathematics 103 (1992) 189-197. We show that a graph admits a topology on its node set which is compatible with the usual connectivity of undirected graphs if, and only if, it is a comparability graph. Then, we give a similar condition for the weak connectivity of oriented graphs and show there is no topology which is compatible with the strong connectivity of oriented graphs. We also give a necessary and sufficient condition for a topology on a discrete set to be 'representable' by an undirected graph. R&sum6 Nous montrons qu'un graphe admet une topologie sur I'ensemble de ses sommets compatible avec la connexit6 usuelle des graphes non-orient& si, et settlement si c'est un graphe de comparabilitt; puis nous donnons une condition similaire pour la connexite faible des graphes orient& et montrons la non-existence d'une topologie compatible avec la connexite forte. Nous donnons Cgalement une condition necessaire et suffisante pour qu'une topologie sur un ensemble discret soit 'representable' par un graphe non-oriente.

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We illustrate how the connections between embedded graphs and delta-matroids can be exploited by using these connections to find an excluded minor characterization and a rough structure theorem for the class of delta-matroids that arise as twists of matroids. Also, we show that several polynomials of embedded graphs, including the Bollobás-Riordan, Penrose, and transition polynomials, are in fact delta-matroidal. . Ralf Rueckriemen was financed by the DFG through grant RU 1731/1-1. 1 8.1. The Penrose polynomial 36 8.2. The Penrose and characteristic polynomials 38 8.3. The transition polynomial 40 References 43 1. Overview

Journal of Intelligent & Fuzzy Systems, 2019

In this article, a technique to construct a new type of topological structures by graphs called topological graphs is introduced. We use the concept of a homeomorphism between topological graphs as a topological property to prove the isomorphic between graphs. We construct a computer program that builds graphs and its topological graphs. This model helps in studying many topological properties such as continuity, connectedness, compactness and separation axioms on graphs. Finally, we present some examples of different types of graphs and their topological graphs and study some of their topological and algebraic properties.

Journal of Combinatorial Theory, Series B, 1979

A matroidal family V is defined to be a collection of graphs such that, for any given graph G, the subgraphs of G isomorphic to a graph in V satisfy the matroid circuit-axioms. Here matroidal families closed under homeomorphism are considered. A theorem of Simks-Pereira shows that when only finite connected graphs are allowed as members of Q, two matroids arise: the cycle matroid and bicircular matroid. Here this theorem is generalized in two directions: the graphs are allowed to be infinite, and they are allowed to be disconnected. In the first case four structures result and in the second case two infinite families of rnatroids are obtained. The main theorem concerns the structures resulting when both restrictions are relaxed simultaneously. We will use standard graph theory terminology as far as possible, as found in [l], [2], or [13]. All graphs will be undirected and possibly infinite, and loops and multiple edges will be allowed. If G is a graph, E(G) denotes the set of edges of G and G\e denotes the graph obtained from G by deleting the edge e. A graph H is homeomorphic from G if it is isomorphic to a graph obtained from G by replacing each edge by a finite path and a graph K is homeomorphic to G if there exists some graph H such that G and K are both homeomorphic from H. The matroid theory terminology will follow [12]. One of the many ways to define a matroid on a finite set is by means of its collection % of circuits, which satisfies the following two axioms: (Cl) No member of V properly contains another.