ABC(T)-graphs: an axiomatic characterization of the median procedure in graphs with connected and G$^2$-connected medians
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| Format: | Preprint |
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2022
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| _version_ | 1866917274828406784 |
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| author | Bénéteau, Laurine Chalopin, Jérémie Chepoi, Victor Vaxès, Yann |
| author_facet | Bénéteau, Laurine Chalopin, Jérémie Chepoi, Victor Vaxès, Yann |
| contents | The median function is a location/consensus function that maps any profile $π$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $π$. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function.
In this paper, we show that modular graphs with $G^2$-connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T$_2$)) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even $Δ$-matroids) are ABCT-graphs and that benzenoid graphs are ABCT$_2$-graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_03587 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | ABC(T)-graphs: an axiomatic characterization of the median procedure in graphs with connected and G$^2$-connected medians Bénéteau, Laurine Chalopin, Jérémie Chepoi, Victor Vaxès, Yann Combinatorics Discrete Mathematics The median function is a location/consensus function that maps any profile $π$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $π$. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function. In this paper, we show that modular graphs with $G^2$-connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T$_2$)) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even $Δ$-matroids) are ABCT-graphs and that benzenoid graphs are ABCT$_2$-graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs. |
| title | ABC(T)-graphs: an axiomatic characterization of the median procedure in graphs with connected and G$^2$-connected medians |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2206.03587 |