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The Path-Distance-Width of Hypercubes

100%
Otachi Y.
Discussiones Mathematicae Graph Theory
|
2013
|
tom 33
|
nr 2
467-470
EN
The path-distance-width of a connected graph G is the minimum integer w satisfying that there is a nonempty subset of S ⊆ V (G) such that the number of the vertices with distance i from S is at most w for any nonnegative integer i. In this note, we determine the path-distance-width of hypercubes.
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Spanning tree congestion of rook's graphs

63%
Kozawa K., Otachi Y.
Discussiones Mathematicae Graph Theory
|
2011
|
tom 31
|
nr 4
753-761
EN
Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T), the congestion of e is the number of edges in G joining the two components of T - e. The congestion of T is the maximum congestion over all edges in T. The spanning tree congestion of G is the minimum congestion over all its spanning trees. In this paper, we determine the spanning tree congestion of the rook's graph Kₘ ☐ Kₙ for any m and n.
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Completely Independent Spanning Trees in (Partial) k-Trees

51%
Matsushita M., Otachi Y., Araki T.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 3
427-437
EN
Two spanning trees T1 and T2 of a graph G are completely independent if, for any two vertices u and v, the paths from u to v in T1 and T2 are internally disjoint. For a graph G, we denote the maximum number of pairwise completely independent spanning trees by cist(G). In this paper, we consider cist(G) when G is a partial k-tree. First we show that [k/2] ≤ cist(G) ≤ k − 1 for any k-tree G. Then we show that for any p ∈ {[k/2], . . . , k − 1}, there exist infinitely many k-trees G such that cist(G) = p. Finally we consider algorithmic aspects for computing cist(G). Using Courcelle’s theorem, we show that there is a linear-time algorithm that computes cist(G) for a partial k-tree, where k is a fixed constant.
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