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1
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Spanning Trees whose Stems have a Bounded Number of Branch Vertices

100%
Yan Z.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 3
773-778
EN
Let T be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of T is denoted by Leaf(T). The subtree T − Leaf(T) of T is called the stem of T and denoted by Stem(T). In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem has a bounded number of branch vertices, and these conditions are best possible.
2
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On a Spanning k-Tree in which Specified Vertices Have Degree Less Than k

100%
Matsumura H.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 1
191-196
EN
A k-tree is a tree with maximum degree at most k. In this paper, we give a degree sum condition for a graph to have a spanning k-tree in which specified vertices have degree less than k. We denote by σk(G) the minimum value of the degree sum of k independent vertices in a graph G. Let k ≥ 3 and s ≥ 0 be integers, and suppose G is a connected graph and σk(G) ≥ |V (G)|+s−1. Then for any s specified vertices, G contains a spanning k-tree in which every specified vertex has degree less than k. The degree condition is sharp.
3
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Minimum vertex ranking spanning tree problem for chordal and proper interval graphs

100%
Dereniowski D.
Discussiones Mathematicae Graph Theory
|
2009
|
tom 29
|
nr 2
253-261
EN
A vertex k-ranking of a simple graph is a coloring of its vertices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et al. proved in [NP-hardness proof and an approximation algorithm for the minimum vertex ranking spanning tree problem, Discrete Appl. Math. 154 (2006) 2402-2410] that the decision problem: given a simple graph G, decide whether there exists a spanning tree T of G such that T has a vertex 4-ranking, is NP-complete. In this paper we improve this result by proving NP-hardness of finding for a given chordal graph its spanning tree having vertex 3-ranking. This bound is the best possible. On the other hand we prove that MVRST problem can be solved in linear time for proper interval graphs.
4
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Closure for spanning trees and distant area

100%
Fujisawa J., Saito A., Schiermeyer I.
Discussiones Mathematicae Graph Theory
|
2011
|
tom 31
|
nr 1
143-159
EN
A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with $deg_G u + deg_G v ≥ n-1$, G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on $deg_G u + deg_G v$ and the structure of the distant area for u and v. We prove that if the distant area contains $K_r$, we can relax the lower bound of $deg_G u + deg_G v$ from n-1 to n-r. And if the distant area itself is a complete graph and G is 2-connected, we can entirely remove the degree sum condition.
5
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Spanning trees with many or few colors in edge-colored graphs

63%
Broersma H., Li X.
Discussiones Mathematicae Graph Theory
|
1997
|
tom 17
|
nr 2
259-269
EN
Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard.
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