# Axiomatic Characterization of the Interval Function of a Bipartite Graph

16 Feb 2017-pp 96-106

TL;DR: A new axiom is introduced: for any x,y,z, R(x,y) = x, y = Rightarrow y in R (x,z) or x in R(y,Z) for any \( x,Y,z \in V\),

Abstract: 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 \(x\in R(y,z)\).

Topics: Bipartite graph (60%), Clique-width (58%), Voltage graph (55%), Edge-transitive graph (55%), Quartic graph (54%)

##### Citations

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TL;DR: The results are surveyed as answers to these questions available from the research papers on the interval function of special graphs using some set of first order axioms defined on an arbitrary transit function.

Abstract: Betweenness is a universal notion present in several disciplines of mathematics. The notion of betweenness has a profound history and many pioneers like Euclid, Pasch, Hilbert have studied betweenness axiomatically. In discrete mathematics too, betweenness is present and several authors have worked on this concept from an axiomatic view. In graph theory, betweenness is developed mainly as metric betweenness, studied using the shortest path metric in a connected graph, thus resulting in the notion of the interval function. Many interesting results are available in graph theory using the interval function. The interval function is generalized to induced path function by replacing shortest paths by induced paths. The induced path betweenness also captured attention among graph theorists with several interesting results to date. From an axiomatic point of view, two pertinent questions can be framed on these functions. Is it possible to axiomatically characterize the interval function of some special graphs using some set of first order axioms defined on an arbitrary transit function? Is it possible to characterize the graphs with the help of their interval functions? In this paper, we survey the results as answers to these questions available from the research papers.

8 citations

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TL;DR: This paper presents characterizations of (claw, paw)-free graphs using axiom (cp) on the standard path transit functions on graphs, namely the interval function, the induced path function, and the all-paths function.

Abstract: The axiomatic study on the interval function, induced path function and all-paths function of a connected graph is a well-known area in metric graph theory and related areas. In this paper, we introduce the following new axiom: (cp) v ∈ R ( u , w ) and v ∈ R ( u , x ) ⇒ w ∈ R ( v , x ) or x ∈ R ( v , w ) , for all distinct u , v , w , x ∈ V . We present characterizations of (claw, paw)-free graphs using axiom (cp) on the standard path transit functions on graphs, namely the interval function, the induced path function, and the all-paths function. We study the underlying graphs of the transit functions which are (claw, paw)-free and Hamiltonian. We present an axiomatic characterization of the interval function on (claw, paw)-free graphs. Furthermore, we obtain an axiomatic characterization of the induced path function on a subclass of (claw, paw)-free graphs.

1 citations

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TL;DR: The class of graphs that are characterized include the important class of Ptolemaic graphs and some proper superclasses of P toleMAic graphs: the distance hereditary graphs and the bridged graphs.

Abstract: 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}.

##### References

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TL;DR: Caratheodory, Helly and Radon type theorems are proved for M-convex sets and the Helly number equals the size of a maximum clique.

Abstract: A set K of vertices in a connected graph is M-convex if and only if for every pair of vertices in K, all vertices of all chordless paths joining them also lie in K. Caratheodory, Helly and Radon type theorems are proved for M-convex sets. The Caratheodory number is 1 for complete graphs and 2 for other graphs. The Helly number equals the size of a maximum clique. The Radon number is one more than the Helly number except possibly for triangle-free graphs, where it is at most 4.

127 citations

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01 Mar 1952-

TL;DR: This paper considers postulates expressed in terms of "segments," "medians," and "betweenness" for trees, lattices, and partially ordered sets.

Abstract: In this paper we consider postulates expressed in terms of "segments," "medians," and "betweenness." Characterizations are obtained for trees, lattices, and partially ordered sets. In general a characterization is given by a system of three postulates. These systems fall in pairs; systems of a pair have two postulates in common. An algebra which has both lattices and trees as special cases is given in the final section.

86 citations

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TL;DR: This work proposes a more general approach for 'path properties' in graphs, focusing on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants.

Abstract: A feasible family of paths in a connected graph G is a family that contains at least one path between any pair of vertices in G. Any feasible path family defines a convexity on G. Well-known instances are: the geodesics, the induced paths, and all paths. We propose a more general approach for such 'path properties'. We survey a number of results from this perspective, and present a number of new results. We focus on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as the clique number and other graph properties.

71 citations

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Abstract: A transit function R on a set V is a function
$$R:VxV \to 2^2 $$
satisfying the axioms
$$u \in R(u,\upsilon ),R(u,\upsilon ) = R(\upsilon ,u)$$
and
$$R(u,u) = \{ u\} $$
, for all
$$u,\upsilon \in V$$
. The all-paths transit function of a connected graph is characterized by transit axioms.

64 citations