Independent metric dimension of Möbius ladders
0
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Let G = (V;E) be a connected graph. The distance d(u; v) between two vertices u; v 2 V is the length of a shortest u- v path. Let W = fw1;w2; : : : ;wkg be a subset of V with an order imposed on it. For v 2 V , the vector r(vjW) = (d(v;w1); d(v;w2); :::; d(v;wk)) is called the metric representation of v with respect to W. If r(vjW) 6= r(ujW) for any two distinct vertices u; v 2 V , then W is called a resolving set of G. The minimum cardinality of a resolving set of G is called the metric dimension of G and is denoted by dim(G) . A subset W of V is called an independent resolving set for G if W is both independent and resolving. The minimum cardinality indim(G) of an independent resolving set in G is called the independent metric dimension of G: In this paper we show that for Mobius ladders, the independent metric dimension and the metric dimension are equal.
Abstract
Let G = (V;E) be a connected graph. The distance d(u; v) between two vertices u; v 2 V is the length of a shortest u- v path. Let W = fw1;w2; : : : ;wkg be a subset of V with an order imposed on it. For v 2 V , the vector r(vjW) = (d(v;w1); d(v;w2); :::; d(v;wk)) is called the metric representation of v with respect to W. If r(vjW) 6= r(ujW) for any two distinct vertices u; v 2 V , then W is called a resolving set of G. The minimum cardinality of a resolving set of G is called the metric dimension of G and is denoted by dim(G) . A subset W of V is called an independent resolving set for G if W is both independent and resolving. The minimum cardinality indim(G) of an independent resolving set in G is called the independent metric dimension of G: In this paper we show that for Mobius ladders, the independent metric dimension and the metric dimension are equal.
참고문헌 (12)
- M. Ali, "On the metric dimension of Mobius Ladders", Ars Combin., 105, pp.403~410 (2012)
- G. Chartrand, "Graphs and Digraphs" (2005), CRC press
- G. Chartrand, "The independent resolving number of a graph", Math. Bohem., 128, pp.379~393 (2003)
- G. Chartrand, "The theory and application of resolvability in graphs: A survey", Congr. Numer., 160, pp.47~68 (2003)
- P.J.B. Chitra, "Resolving sets without isolated vertices", Procedia Computer Science, 74, pp.38~42 (2015)
- F. Harary, "On the metric dimension of a graph", Ars Combin., 318, pp.191~195 (1976)
- M. Munir, "Metric dimension of the Mobius Ladder", Ars Combin.
- V. Saenpholphat, "Some results on connected resolvability in graphs", Congr. Numer., 158, pp.5~19 (2002)
- V. Saenpholphat, "On connected resolvability of graphs", Australas. J. Combin., 28, pp.25~37 (2003)
- V. Saenpholphat, "Conditional resolvability in graphs : A survey", Int. J. Math. Math. Sci, 2004, pp.1997~2017 (2004)
- V. Saenpholphat, "Connected resolvability of graphs", Czechoslovak Math. J, 53, pp.827~840 (2003)
- P. J. Slater, "Leaves of trees", Congr. Numer, 14, pp.549~559 (1975)