Maximum induced trees in graphs

P. Erdős, Michael Saks, Vera T. Sós

Research output: Contribution to journalArticle

67 Citations (Scopus)

Abstract

Let t(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t(G) to other parameters associated with G: the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity. The central result is a set of upper and lower bounds for the function f(n, ρ{variant}), defined to be the minimum of t(G) over all connected graphs with n vertices and n - 1′ + ρ{variant} edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f(n, ρ{variant}) is surprisingly small; in particular f(n, cn) = 2 loglogn + O(logloglogn) for any constant c > 0, and f(n, n1 + γ) = 2 log(1 + 1 γ) ± 4 for 0 <γ <1 and n sufficiently large. Bounds on t(G) are obtained in terms of the size of the largest clique. These are used to formulate bounds for a Ramsey-type function, N(k, t), the smallest integer so that every connected graph on N(k, t) vertices has either a clique of size k or an induced tree of size t. Tight bounds for t(G) from the independence number α(G) are also proved. It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2r - 1 vertices.

Original languageEnglish
Pages (from-to)61-79
Number of pages19
JournalJournal of Combinatorial Theory. Series B
Volume41
Issue number1
DOIs
Publication statusPublished - 1986

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Connected graph
Graph in graph theory
Independence number
Clique
Radius
Maximum Clique
Upper and Lower Bounds
Connectivity
Path
Integer
Subset
Vertex of a graph
Range of data
Relationships

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Maximum induced trees in graphs. / Erdős, P.; Saks, Michael; Sós, Vera T.

In: Journal of Combinatorial Theory. Series B, Vol. 41, No. 1, 1986, p. 61-79.

Research output: Contribution to journalArticle

Erdős, P. ; Saks, Michael ; Sós, Vera T. / Maximum induced trees in graphs. In: Journal of Combinatorial Theory. Series B. 1986 ; Vol. 41, No. 1. pp. 61-79.
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