# On Trees of Bounded Degree with Maximal Number of Greatest Independent Sets

On Trees of Bounded Degree with Maximal Number of Greatest Independent Sets Given n and d, we describe the structure of trees with the maximal possible number of greatest independent sets in the class of n-vertex trees of vertex degree at most d.We show that the extremal tree is unique for all even n but uniqueness may fail for odd n; moreover, for d = 3 and every odd n ≥ 7, there are exactly ⌈(n − 3)/4⌉ + 1 extremal trees. In the paper, the problem of searching for extremal (n, d)-trees is also considered for the 2-caterpillars; i.e., the trees in which every vertex lies at distance at most 2 from some simple path. Given n and d ∈ {3, 4}, we completely reveal all extremal 2-caterpillars on n vertices each of which has degree at most d. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Industrial Mathematics Springer Journals

# On Trees of Bounded Degree with Maximal Number of Greatest Independent Sets

, Volume 12 (2) – May 29, 2018
13 pages

/lp/springer_journal/on-trees-of-bounded-degree-with-maximal-number-of-greatest-independent-RY3SSbNr8s
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1990-4789
eISSN
1990-4797
D.O.I.
10.1134/S1990478918020175
Publisher site
See Article on Publisher Site

### Abstract

Given n and d, we describe the structure of trees with the maximal possible number of greatest independent sets in the class of n-vertex trees of vertex degree at most d.We show that the extremal tree is unique for all even n but uniqueness may fail for odd n; moreover, for d = 3 and every odd n ≥ 7, there are exactly ⌈(n − 3)/4⌉ + 1 extremal trees. In the paper, the problem of searching for extremal (n, d)-trees is also considered for the 2-caterpillars; i.e., the trees in which every vertex lies at distance at most 2 from some simple path. Given n and d ∈ {3, 4}, we completely reveal all extremal 2-caterpillars on n vertices each of which has degree at most d.

### Journal

Journal of Applied and Industrial MathematicsSpringer Journals

Published: May 29, 2018

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