Anchored Hyperspaces and Multigraphs
DOI:
https://doi.org/10.11575/cdm.v14i1.62646Abstract
Consider a multigraph $X$ as a metric space and $p \in X$. The anchored hyperspace at $p$ is the set
$C_p(X) =$ {$A \subseteq X : p \in A, A$ connected and compact}.
In this paper we will prove that $C_p(X)$ is a polytope if in this set is considered the Hausdorff's metric $H$. Further we will show that, if $X$ is a locally connected compact metric space such that $C_p(X)$ is a polytope for each $p \in X$, then $X$ must be a multigraph.
References
A. Barvinok, I Novik, A centrally symmetric version of the cyclic polytope, Discrete Comput. Geom.: 39(1-3) (2008), 7699.
A. Barvinok, S. J. Lee, I Novik, Explicit constructions of centrally symmetric k−neighborly polytopes and large strictly antipodal sets, Discrete Comput. Geom.: 49(3) (2013), 429443.
S. Bermudo, J. M. Rodríguez and José M. Sigarreta, Gromov hyperbolic graphs, Discrete Mathematics, 313 (2013),
1575-1585.
D. Bryant, B. R. Smith, Descompositions of complete multigraphs into cycles of varying lenght, Journal of Combinatorial Theory: 129 (2018), 79106.
R.Duda, On the hyperspace of subcontinua of a nite graph I, Fund. Math. 62 (1968), pp.265-286.
J. Dugundji, Topology Allyn and Bacon Series in Advanced Mathematics 1976.
M. Gentner, D. Rautenbach Feedback vertex sets in cubic multigraphs, Journal Discrete Mathematics., 338 (12) (2015) pp. 21792185.
A. Illanes, Cells and cubes in hyperspaces, Fund. Math., 130 (1988) pp. 57-65.
S. Klee, E. Nevo, I. Novik, H Zheng, A lower bound theorem for centrally symmetric simplicial polytopes, Discrete Comput. Geom.: to appear (2018); DOI 10.1007/s00454-018-9978-z.
José M. Sigarreta, Hyperbolicity in median graphs, Proceedings Mathematical Sciences, 123 (4) (2013), 455-467.
S.B. Nadler, Locating cones and Hilbert cubes in hyperspaces. Fun. Math., 79 (1973), 233250.
S. B. Nadler Jr., Hyperspaces of sets. A Text with Research Questions. 33 Marcel Dekker, Inc. 1978.
I. Novik, A tale of centrally symmetric polytopes and spheres, Departament of Mathematics. University of Washington
Seattle 2018.
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