Preface.- 1 Graphs.- 2 Bipartite Graphs.- 3 Cubes.- 4 Cubical Graphs.- 5 Partial Cubes.- 6 Lattice Embeddings.- 7 Hyperplane Arrangements.- 8 Token Systems.- Notation.- References.- Index.

Graphs and Cubes

Betweenness in graphs: A short survey on shortest and induced path betweenness

The axiomatic approach with the interval function and induced path transit function of a connected graph is an interesting topic in metric and related graph theory. In this paper, we introduce a new axiom:


(bp) for any $ x,y,z \in V$, $R(x,y)=\{x,y\} \Rightarrow y\in R(x,z)$ or $x\in R(y,z)$.

Axiomatic Characterization of the Interval Function of a Bipartite Graph

Axiomatic characterization of the interval function of a bipartite graph

Transit functions provide a unified approach to study notions of intervals, convexities, and betweenness. Recently, their scope has been extended to certain set systems associated with clustering. We characterize here the class of set systems that correspond to k-ary monotonic transit functions. Convexities form a subclass and are characterized in terms of transit functions by two additional axioms. We then focus on axiom systems associated with weak hierarchies as well as other generalizations of hierarchical set systems.

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Axiomatic characterization of transit functions of weak hierarchies

$k$-point crossover operators and their recombination sets are studied from different perspectives. We show that transit functions of $k$-point crossover generate, for all $k>1$, the same convexity as the interval function of the underlying graph. This settles in the negative an open problem by Mulder about whether the geodesic convexity of a connected graph $G$ is uniquely determined by its interval function $I$. The conjecture of Gitchoff and Wagner that for each transit set $R_k(x,y)$ distinct from a hypercube there is a unique pair of parents from which it is generated is settled affirmatively. Along the way we characterize transit functions whose underlying graphs are Hamming graphs, and those with underlying partial cube graphs. For general values of $k$ it is shown that the transit sets of $k$-point crossover operators are the subsets with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of $k$-point crossover on binary strings form topes of uniform oriented matroid of VC-dimension $k+1$. The Topological Representation Theorem for oriented matroids therefore implies that $k$-point crossover operators can be represented by pseudosphere arrangements. This provides the tools necessary to study the special case $k=2$ in detail.

Topological Representation of the Transit Sets of k-Point Crossover Operators

The axiomatic approach with the interval function, induced path transit function and all-paths transit function of a connected graph form a well studied area in metric and related graph theory. In this paper we introduce the first order axiom:

cp For any pairwise distinct vertices $$a,b,c,d\in V$$

$$b \in Ra,c$$ and $$b \in Ra,d\Rightarrow c\in Rb,d$$ or $$d\in Rb,c$$.

We study this new axiom on the interval function, induced path transit function and all-paths transit function of a connected simple and finite graph. We present characterizations of claw and paw-free graphs using this axiom on standard path transit functions on graphs, namely the interval function, induced path transit function and the all-paths transit function. The family of 2-connected graphs for which the axiom cp is satisfied on the interval function and the induced path transit function are Hamiltonian. Additionally, we study arbitrary transit functions whose underlying graphs are Hamiltonian.

Axiomatic Characterization of Claw and Paw-Free Graphs Using Graph Transit Functions

Transit functions provide a unifying approach to many results on intervals, convexities, and betweenness. Here we show that hierarchical structures arising in cluster analysis and phylogenetics have a natural characterization in terms of transit functions and that hierarchies are identified by multiple combinations of independent axioms.

/pdf/axiomatic-characterization-of-transit-functions-of-2joh3n1zbg.pdf

Ferdoos Hossein Nezhad

Papers

Topological Representation of the Transit Sets of k-Point Crossover Operators

Axiomatic Characterization of Claw and Paw-Free Graphs Using Graph Transit Functions

Axiomatic characterization of transit functions of hierarchies

Axiomatic Characterization of the Interval Function of a Bipartite Graph

Axiomatic characterization of the interval function of a bipartite graph