Sharp threshold for embedding balanced spanning trees in random geometric graphs

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph G(n,r,d) ${\mathscr{G}}(n,r,d)$. In particular, we find the sharp...

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Veröffentlicht in:Journal of graph theory Jg. 107; H. 1; S. 107 - 125
Hauptverfasser: Espuny Díaz, Alberto, Lichev, Lyuben, Mitsche, Dieter, Wesolek, Alexandra
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Hoboken Wiley Subscription Services, Inc 01.09.2024
Wiley
Schlagworte:
Algorithms
Graph theory
Graphs
linear‐time algorithm
Mathematics
Polynomials
random geometric graphs
sharp threshold
spanning trees
Trees (mathematics)
Vertex sets
ISSN:0364-9024, 1097-0118
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Zusammenfassung:A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph G(n,r,d) ${\mathscr{G}}(n,r,d)$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.
Bibliographie:ObjectType-Article-1
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ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.23106