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 \(x\in R(y,z)\).
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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.
13 citations
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.
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}.
1 citations
References
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28 Jun 2006
TL;DR: This paper addresses the question what kind of restrictions could be imposed to obtain axiomatic characterizations of J by transit axioms only and restricts ourselves to functions J that satisfy betweenness, or monotonicity.
Abstract: textThe 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 $u \
otin J(w, v)$ 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.
17 citations
TL;DR: It turns out that the class of graphs for which I G satisfies is a proper subclass of distance hereditary graphs and the class for which J G satisfies (ii) is a Proper superclass ofdistance hereditary graphs.
12 citations
TL;DR: From this, two new characterizations of the interval function of a block graph using axioms on an arbitrary transit function R are deduced.
11 citations
TL;DR: In this paper, the exact degree of approximation for complex bivariate Balazs-Szabados operators of tensor-product kind is obtained and a Voronovskaja-type theorem of these operators is given.
Abstract: This study deals with approximation properties by the complex bivariate Balazs-Szabados operators of tensor-product kind. The upper and lower estimates and a Voronovskaja-type theorem of these operators are given. The exact degree of approximation for these operators is obtained.
11 citations
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TL;DR: In this article, a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree.
Abstract: We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.
10 citations