On The Strong Metric Dimension of Sun Graph,
Windmill Graph, and M¨
obius Ladder Graph
Mila Widyaningrum and Tri Atmojo Kusmayadi
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret, Surakarta, Indonesia
E-mail: [email protected], [email protected]
Abstract. LetGbe a connected graph with vertex setV(G) and edge setE(G). The interval I[u, v] betweenuand vis defined as the collection of all vertices that belong to some shortest u-v path. A vertexs ∈V(G) strongly resolves two verticesuandvif ubelongs to a shortest v-spath orvbelongs to a shortestu-s path. A setS ⊂V(G) is a strong resolving set if every two distinct vertices ofGare strongly resolved by some vertex ofS. The smallest cardinality of strong resolving set is called a strong metric basis. The strong metric dimension ofG, denoted bysdim(G), is defined as the cardinality of the strong metric basis. In this paper we determine the strong metric dimension of a sun graph Sn, a windmill graph K
m
n, and a M¨obius ladder
graphMn. We obtain the strong metric dimension of sun graph Sn isn - 1 for n ≥3. The
strong metric dimension of windmill graphKm
n is (n- 1)m- 1 form≥2 andn≥3. The strong
metric dimension of M¨obius ladder graphMnwithn≥5 is 2⌈ n+2
4 ⌉forneven.
1. Introduction
Seb¨o and Tannier [8] introduced the concept of strong metric dimension in 2004. In 2007 the strong metric dimension of graph and digraph was introduced by Oellermann and Peters-Fransen [7]. LetGbe a connected graph with vertex setV(G) and edge setE(G), Oellermann and Peters-Fransen [7] defined for two vertices u, v∈V(G), the interval I[u, v] between u and v to be the collection of all vertices that belong to some shortest u - v path. If u belongs to a shortestv-s
path, denoted by u ∈ I[v, s], or v belongs to a shortest u-s path, denoted by v ∈I[u, s] then a vertex s strongly resolves two vertices u and v. A set of verticesS ⊂V(G) is strong resolving set for G if every two distinct vertices of G are strongly resolved by some vertex of S. The smallest cardinality of strong resolving set is called strong metric basis. The cardinality of the strong metric basis is a strong metric dimension of G, denoted bysdim(G).
Some authors have investigated the strong metric dimension to some graph classes. In 2004 Seb¨o and Tannier [8] observed the strong metric dimension of complete graphKn, cycle graph
Cn, and tree. Kratica [3] determined the strong metric dimension of hamming graph Hn,k in
2012. At the same year, Kratica [4] already observed the strong metric dimension of some convex polytope graph. In 2013 Yi [9] determined that the metric dimension of G is 1 if and only ifG
is a path Pn and the strong metric dimension of G is n - 1 if and only if G is complete graph
Kn. Kusmayadi et al. [5] determined the strong metric dimension of some related wheel graph
such as sunflower graph, t-fold wheel graph, helm graph, and friendship graph in 2016. In this paper we determine the strong metric dimension of a sun graph Sn, a windmill graphKnm, and
2. Main Results
2.1. Strong Metric Dimension
Let G be a connected graph with a set of vertices V(G) and a set of edges E(G). Let S be a subset of V(G). Oellermann and Peters-Fransen [7] defined for two vertices u, v ∈ V(G), the intervalI[u, v] betweenu andv to be the collection of all vertices that belong to some shortest
u −vpath. A vertexs∈S strongly resolves two vertices uandv ifu∈I[v, s] orv ∈I[u, s]. A vertex set S of Gis a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. The smallest cardinality of strong resolving set is called strong metric basis of G. The strong metric dimension of a graph G is defined as the cardinality of strong metric basis denoted by sdim(G). We often make use of the following lemma and properties about strong metric dimension given by Kratica et al. [4].
Lemma 2.1 Let u, v ∈V(G), u ̸= v,
(i) d(w,v) ≤ d(u,v) for each w such that uw∈ E(G), and (ii) d(u,w) ≤ d(u,v) for each w such that vw∈ E(G).
Then there does not exist vertex a ∈ V(G), a ̸= u,v that strongly resolves vertices u and v.
Property 2.1 If S ⊂V(G) is strong resolving set of graph G, then for every two vertices
u, v∈V(G) satisfying conditions1 and 2 of Lemma 2.1, obtained u ∈ S or v ∈ S.
Property 2.2 If S ⊂V(G) is strong resolving set of graph G, then for every two vertices
u, v∈V(G) satisfying d(u,v) = diam(G), obtained u ∈ S or v∈ S.
2.2. The Strong Metric Dimension of Sun Graph
Mirzakhah and Kiani [6] defined the sun graph Sn forn≥3 as a graph obtained by connecting
each vertex of cycleCnto a pendant vertex with an edge. The sun graphSn can be depicted as
in Figure 1.
u
2
u
3
u
4
u
1
u
n
v
2
v
3
v
4
v
n
v
1
Figure 1. Sun graph Sn
Theorem 2.1 Let Sn be the sun graph with n ≥ 3. Then sdim(Sn) = n − 1.
Proof. First, we prove that for every integer n≥3, ifS is a strong resolving set of sun graphSn
then |S | ≥n −1. We know that S is a strong resolving set of sun graph Sn. Suppose that S
contains at mostn- 2 vertices, then|S |< n- 1. LetV1, V2 ⊂V(Sn), with V1 ={v1, v2, . . . , vn}
and V2 = {u1, u2, . . . , un}. Now, we define S1 = V1 ∩S and S2 = V2 ∩S. Without loss of
are two distinct verticesvi and vj ∈V1\S1, wherei̸=j such that for everys∈S,vi ∈/ I[vj, s]
and vj ∈/ I[vi, s]. This contradicts with the supposition that S is a strong resolving set. Thus |S | ≥n- 1.
Then, we prove that for every integer n ≥ 3, if S = {v1, v2, . . . , vn−1} then S is a strong
resolving set of sun graph Sn. We prove that for every two distinct verticesu, v ∈ V(Sn) \ S
where u̸= v is strongly resolved by a vertexs∈ S. There are two possible pairs of vertices. (i) A pair of vertices (ui, vn) withi= 1, 2,. . .,n.
For every integer 1 ≤ i ≤ ⌈n−21⌉, we obtain the shortest path between v⌈n−1 2 ⌉ and
vn is v⌈n−1 2 ⌉, u⌈
n−1 2 ⌉, u⌈
n−1
2 ⌉−1, . . . , u1, un, vn. So that ui ∈ I[v⌈ n−1
2 ⌉, vn]. Then for
every integer ⌊n+12 ⌋ ≤ i ≤ n, we obtain the shortest path between v⌊n+1
2 ⌋ and vn is
v⌊n+1 2 ⌋, u⌊
n+1 2 ⌋, u⌊
n+1
2 ⌋+1, . . . , un−1, un, vn. So thatui ∈I[v⌊ n+1
2 ⌋, vn].
(ii) A pair of vertices (ui, uj) withi, j = 1, 2, . . .,nand i̸=j.
For every integer i, j ∈ [1, n] without loss of generality we may assume 1 ≤ i < j ≤ n, therefore ui anduj will be strongly resolved byvi so that ui ∈I[vi, uj].
From every possible pairs of vertices, there exists a vertex vi ∈ S with i ∈ [1, n−1] which
strongly resolves two distinct vertices of V(Sn) \ S. Thus S is a strong resolving set of sun
graph Sn.
We obtain a setS ={v1, v2, . . . , vn−1} is a strong resolving set of sun graph Sn withn ≥3
and we have|S | ≥n - 1, so thatS is a strong metric basis of sun graph Sn. Hencesdim(Sn)
=n - 1. ⊓⊔
2.3. The Strong Metric Dimension of Windmill Graph
Gallian [1] defined the windmill graph Km
n is a graph consists of m copies ofKn with a vertex
in common. The windmill graph Km
n can be depicted as in Figure 2.
v
1v
2v
n - 2
v
n - 1v
nv
n + 1v
2n - 3v
(n -1) 2v
(n -1)(m -1) + 1v
(n -1)(m -1) + 2v
(n -1)m - 1v
(n -1)mc
Figure 2. Windmill graph Km n
Theorem 2.2 Let Km
n be the windmill graph with m ≥ 2 and n ≥ 3. Then sdim(Knm) =
(n−1)m−1.
Proof. First, we prove that for every integer m ≥ 2 andn ≥3, if S is a strong resolving set of windmill graph Km
n then |S| ≥(n- 1)m - 1. Consider a pair of vertices (vi, vj) for i, j = 1, 2,
we obtainvi ∈S orvj ∈S. It meansS has at least one vertex from setsXij ={vi, vj}withi, j
= 1, 2,. . ., (n−1)mand i≠ j. The minimum number of vertices from setsXij is (n- 1)m- 1.
Therefore, |S | ≥(n−1)m−1.
After that, we prove that for every integerm ≥2 and n≥ 3, ifS = {v1, v2, . . . , v(n−1)m−1}
then S is a strong resolving set of Km
n graph. For every two distinct vertices u, v∈ V(Knm)\S,
there is a vertex s ∈ S which strongly resolves u and v. Let us prove that a pair of vertices (c, v(n−1)m) is strongly resolved byvifori= 1, 2,. . ., (n−1)(m−1). The shortest path between
2.4. The Strong Metric Dimension of M¨obius Ladder Graph
Guy and Harary [2] defined the M¨obius ladder graph Mn is a graph constructed by connecting
vertices u and v in the cycle Cn if d(u,v) = diam(Cn). The M¨obius ladder graph Mn can be
depicted as in Figure 3 and Figure 4.
v
4Figure 3. M¨obius ladder graphMn forneven
v__
n+7Figure 4. M¨obius ladder graphMn fornodd
Theorem 2.3 Let Mn be the M¨obius ladder graph withn ≥5. Then sdim(Mn) =2 ⌈n+24 ⌉ for
n even.
Proof. First, we prove that for everyn≥5, ifS is a strong resolving set of M¨obius ladder graph
graph Mn. Suppose that S contains at most 2 ⌈n+24 ⌉ − 1 vertices, then | S |< 2 ⌈n+24 ⌉. Let
4 ⌉. Otherwise, there are two distinct verticesvi and vj, where
vi ∈ V1\S1 and vj ∈ V2\S2 such that for every s ∈ S we obtain vi ∈/ I[vj, s] and vj ∈/ I[vi, s].
This contradicts with the supposition that S is a strong resolving set. Thus |S| ≥2 ⌈n+2 4 ⌉.
4 ⌉}is a strong resolving set of M¨obius ladder graphMnforneven. We prove that for
every two distinct verticesvi, vj ∈V(Mn)\S, wherei, j= (⌈n+24 ⌉, n2]∪(n2+⌈n+24 ⌉,n] is strongly
resolved by a vertex s∈ S. There are three possible pairs of vertices. (i) A pair of vertices (vi, vj) where ⌈n+24 ⌉ < i, j ≤ n2,i̸=j.
vj will be strongly resolved by a vertex vn 2+⌈
n+2 4 ⌉ if
n
2 +⌈n+24 ⌉ < j ≤i+ n2, we obtain the
shortest path between vi and vn 2+⌈
From every possible pairs of vertices, there exists a vertex s ∈ S which strongly resolves two distinct vertices ofV(Mn) \ S. ThusS is a strong resolving set of M¨obius ladder graph Mn.
According to the discussion above, we have the strong metric dimension of a sun graph Sn,
a windmill graph Km
n , and a M¨obius ladder graph Mn for n even. Furthermore, we conclude
this paper with the following open problem for the direction of further research which is still in progress.
Open Problem: Determine the strong metric dimension of a M¨obius ladder graph Mn for n
odd.
Acknowledgments
The authors gratefully acknowledge the support from Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret, Surakarta. Then, we wish to thank the referees for their suggestions which helped to improve the paper.
References
[1] Gallian, Joseph A., A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics (2016), ♯(DS6).
[3] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. ˇCangalovi´c,Minimal Doubly Resolving Sets and The Strong Metric Dimension of Hamming Graphs, Applicable Analysis and Discrete Mathematics6(2012), 63–71.
[4] Kratica, J., V. Kovaˇcevi´c-Vujˇci´c, and M. ˇCangalovi´c,Minimal Doubly Resolving Sets and The Strong Metric Dimension of Some Convex Polytope, Applied Mathematics and Computation218(2012), 9790–9801. [5] Kusmayadi, T. A., S. Kuntari, D. Rahmadi, and F. A. Lathifah,On The Strong Metric Dimension of Some
Related Wheel Graph, Far East Journal of Mathematical Sciences (FJMS)99(2016), no. 9, 1325–1334. [6] Mirzakhah, M. and D. Kiani, The Sun Graph is Determined by Its Signless Laplacian Spectrum, Electronic
Journal of Linear Algebra20(2010), 610–620.
[7] Oellermann, O. and J. Peters-Fransen,The Strong Metric Dimension of Graphs and Digraphs, Discrete Applied Mathematics155(2007), 356–364.
[8] Seb¨o, A. and E. Tannier, On Metric Generators of Graph, Mathematics and Operations Research 29(2)
(2004), 383–393.
