On the decomposition dimension of trees

Enomoto, Hikoe; Nakamigawa, Tomoki

doi:10.1016/s0012-365x(01)00454-x

Cited by 9 publications

(13 citation statements)

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“…The decomposition dimensions of trees that are not paths have been studied in [7,26], where bounds for them have been determined. However, there is no general formula for the decomposition dimension of a tree that is not a path.…”

Section: Connected Resolving Decompositions a Resolving Decompositionmentioning

confidence: 99%

“…Resolving sets in graphs have been studied further in [1,2,3,5,6,8,14,20,21,24,34,37,38]. Recently, these concepts have been extended in various ways and studied for different subjects in graph theory, including such diverse aspects as the partition of the vertex set, decomposition, orientation, domination, coloring in graphs [15,16,17,19,23,25,35,36,39,47,51], and others [22]. Many invariants arising from the study of resolving sets in graph theory offer subjects for applicable research.…”

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confidence: 99%

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Conditional resolvability in graphs: a survey

Saenpholphat

Zhang

2004

International Journal of Mathematics and Mathematical Sciences

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For an ordered set W={w1,w2,Ã¢Â€Â¦,wk} of vertices and a vertex v in a connected graph G, the code of v with respect to W is the k-vector cW(v)=(d(v,w1),d(v,w2),Ã¢Â€Â¦,d(v,wk)), where d(x,y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct codes with respect to W. The minimum cardinality of a resolving set for G is its dimension dim(G). Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition and coloring in graphs, or by combining resolving property with another graph-theoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs

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Section: Connected Resolving Decompositions a Resolving Decompositionmentioning

confidence: 99%

mentioning

confidence: 99%

Conditional resolvability in graphs: a survey

Saenpholphat

Zhang

2004

International Journal of Mathematics and Mathematical Sciences

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“…Recently, these concepts were rediscovered by Johnson [5], [6] of the Pharmacia Company while attempting to develop a capability of large datasets of chemical graphs. Resolving decompositions in graphs were first introduced and studied in [1] and further studied in [3], [4]. The connected resolving decompositions in graph have been studied in [13].…”

Section: D(e F )mentioning

confidence: 99%

“…), it follows that D is a resolving decomposition of G. Subcase 2.5 : G contains a subgraph that is isomorphic to A 5 . Since l = 4, it follows that v 1 w, vv 2 , vv 4 / ∈ E(G). If vv 3 ∈ E(G), then G contains a subgraph that is isomorphic to A 1 , and so the result follows by Subcase 2.1.…”

Section: Characterizing Graphs With Connected Decomposition Number M −mentioning

confidence: 99%

On Connected Resolving Decompositions in Graphs

Saenpholphat

Zhang

2004

Czechoslovak Mathematical Journal

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“…Enomoto [4] proved that dec(G) ≤ n − 1 for every n-vertex graph G except complete graphs with 3 or 4 vertices. Enomoto and Nakamigawa [5] proved another lower bound in terms of the maximum degree ∆(G): (2) dec(G) ≥ lg ∆(G) + 1, where lg denotes log 2 (we also use ln for log e ). They proved that for every r ≥ 1 and k ≥ 3 there is a tree T with ∆(T ) = k and dec(T ) = lg k + r. For graphs with small diameter, the bound in (1) is sharper, and we will study graphs with this property.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Methods for Decomposition Dimension of Graphs

Kündgen

West

2003

Graphs and Combinatorics

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Abstract. In a graph G, the distance from an edge e to a set F ⊆ E(G) is the vertex distance from e to F in the line graph L(G). For a decomposition of E(G) into k sets, the distance vector of e is the k-tuple of distances from e to these sets. The decomposition dimension dec(G) of G is the smallest k such that G has a decomposition into k sets so that the distance vectors of the edges are distinct.For the complete graph K n and the k-dimensional hypercube Q k , we prove (2The upper bounds use probabilistic methods directly or indirectly. We also prove that random graphs with edge probability p such that pn 1−ε → ∞ for some positive constant ε have decomposition dimension Θ(ln n) with high probability.

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On the decomposition dimension of trees

Cited by 9 publications

References 0 publications

Conditional resolvability in graphs: a survey

Conditional resolvability in graphs: a survey

On Connected Resolving Decompositions in Graphs

Probabilistic Methods for Decomposition Dimension of Graphs

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