The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \\in J(u, v)$ implies $u \\notin J(w, v)$ and $x \\in J(u, v)$ implies $J(u, x \\subseteq J(u, v)$, and it is monotone if $x, y \\in J(u, v)$ implies $J(x, y) \\subseteq J(u, v)$. The induced path function of aconnected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.betweenness;induced path;transit function;monotone;house domino;long cycle;p-graph

The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \\in J(u, v)$ implies $u \\notin J(w, v)$ and $x \\in J(u, v)$ implies $J(u, x \\subseteq J(u, v)$, and it is monotone if $x, y \\in J(u, v)$ implies $J(x, y) \\subseteq J(u, v)$. The induced path function of a connected graph satisfying the betweenness and monotone axioms are characterized by transit axioms

The Toll Walk Transit Function of a Graph: Axiomatic Characterizations and First-Order Non-definability

A walk $W=w_1w_2\dots w_k$, $k\geq 2$, is called a toll walk if $w_1\neq w_k$ and $w_2$ and $w_{k-1}$ are the only neighbors of $w_1$ and $w_k$, respectively, on $W$ in a graph $G$. A toll walk interval $T(u,v)$, $u,v\in V(G)$, contains all the vertices that belong to a toll walk between $u$ and $v$. The toll walk intervals yield a toll walk transit function $T:V(G)\times V(G)\rightarrow 2^{V(G)}$. We represent several axioms that characterize the toll walk transit function among chordal graphs, trees, asteroidal triple-free graphs, Ptolemaic graphs, and distance hereditary graphs. We also show that the toll walk transit function can not be described in the language of first-order logic for an arbitrary graph.Comment: 31 pages, 4 figures, 25 reference

A Characterization of Uniquely Representable Graphs

The betweenness structure of a finite metric space M =(X, d) is a pair ℬ (M)=(X, βM) where βM is the so-called betweenness relation of M that consists of point triplets (x, y, z) such that d(x, z)= d(x, y)+ d(y, z). The underlying graph of a betweenness structure ℬ =(X, β)isthe simple graph G(ℬ)=(X, E) where the edges are pairs of distinct points with no third point between them. A connected graph G is uniquely representable if there exists a unique metric betweenness structure with underlying graph G. It was implied by previous works that trees are uniquely representable. In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs. Further, we prove that two related classes of graphs coincide with the class of block graphs and the class of distance-hereditary graphs, respectively. We show that our results hold not only for metric but also for almost-metric betweenness structures. © 2021 Péter G.N. Szabó

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.

Geodetic Graphs and Convexity.

A graph is geodetic if each two vertices are joined by a unique shortest path. The problem of characterizing such graphs was posed by Ore in 1962; although the geodetic graphs of diameter two have been described and classified by Stemple and Kantor, little is known of the structure of geodetic graphs in general. In this work, geodetic graphs are studied in the context of convexity in graphs: for a suitable family (PI) of paths in a graph G, an induced subgraph H of G is defined to be (PI)-convex if the vertex-set of H includes all vertices of G lying on paths in (PI) joining two vertices of H. Then G is (PI)-geodetic if each (PI)-convex hull of two vertices is a path. For the family (GAMMA) of geodesics (shortest paths) in G, the (GAMMA)-geodetic graphs are exactly the geodetic graphs of the original definition. For various families (PI), the (PI)-geodetic graphs are characterized. The central results concern the family (UPSILON) of chordless paths of length no greater than the diameter; the (UPSILON)-geodetic graphs are called ultrageodetic. For graphs of diameter one or two, the ultrageodetic graphs are exactly the geodetic graphs. A geometry (P,L,F) consists of an arbitrary set P, an arbitrary set L, and a set F (L-HOOK EQ) P x L. The point-flag graph of a geometry is defined here to be the graph with vertex-set P (UNION) F whose edges are the pairs {p,(p,1)} and {(p,1),(q,1)} with p,q (ELEM) P, 1 (ELEM) L, and (p,1),(q,1) (ELEM) F. With the aid of the Feit-Higman theorem on the nonexistence of generalized polygons and the collected results of Fuglister, Damerell-Georgiacodis, and Damerell on the nonexistence of Moore geometries, it is shown that two-connected ultrageodetic graphs of diameter greater than two are precisely the graphs obtained via the subdivision, with a constant number of new vertices, either of all of the edges incident with a single vertex in a complete graph, or of all edges of the form {p,(p,1)} in the point-flag graph of a finite projective plane

Bucolic Complexes

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.Comment: 45 pages, 4 figure