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## Discussiones Mathematicae Graph Theory

2002 | 22 | 2 | 305-323
Tytuł artykułu

### Connected partition dimensions of graphs

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = min{d(v,x)|x ∈ S}. For an ordered k-partition Π = {S₁,S₂,...,Sₖ} of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S₁), d(v,S₂),..., d(v,Sₖ)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = {S₁,S₂,...,Sₖ} of V(G) is connected if each subgraph $⟨S_i⟩$ induced by $S_i$ (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V(G) is the connected partition dimension cpd(G) of G. Thus 2 ≤ pd (G) ≤ cpd(G) ≤ n for every connected graph G of order n ≥ 2. The connected partition dimensions of several classes of well-known graphs are determined. It is shown that for every pair a, b of integers with 3 ≤ a ≤ b ≤ 2a-1, there is a connected graph G having pd(G) = a and cpd(G) = b. Connected graphs of order n ≥ 3 having connected partition dimension 2, n, or n-1 are characterized.
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EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
305-323
Opis fizyczny
Daty
wydano
2002
otrzymano
2001-04-24
poprawiono
2001-10-20
Twórcy
• Department of Mathematics and Statistics, Western Michigan University, Kalamozoo, MI 49008, USA
autor
• Department of Mathematics and Statistics, Western Michigan University, Kalamozoo, MI 49008, USA
Bibliografia
• [1] G. Chartrand and L. Lesniak, Graphs & Digraphs, third edition (Chapman & Hall, New York, 1996).
• [2] G. Chartrand, C. Poisson and P. Zhang, Resolvability and the upper dimension of graphs, Inter. J. Comput. Math. Appl. 39 (2000) 19-28, doi: 10.1016/S0898-1221(00)00126-7.
• [3] G. Chartrand, E. Salehi and P. Zhang, On the partition dimension of a graph, Congress. Numer. 131 (1998) 55-66.
• [4] G. Chartrand, E. Salehi and P. Zhang, The partition dimension of a graph, Aequationes Math. 59 (2000) 45-54, doi: 10.1007/PL00000127.
• [5] F. Harary and R.A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976) 191-195.
• [6] M.A. Johnson, Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharm. Statist. 3 (1993) 203-236, doi: 10.1080/10543409308835060.
• [7] M.A. Johnson, Browsable structure-activity datasets, preprint.
• [8] P.J. Slater, Leaves of trees, Congress. Numer. 14 (1975) 549-559.
• [9] P.J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988) 445-455.
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