- Email: [email protected]
Vertices, edges, distances and metric dimension in graphs
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 55 (2016) 191–194 www.elsevier.com/locate/endm
Vertices, edges, distances and metric dimension in graphs Ismael Gonz´alez Yero 1 Departamento de Matem´ aticas, Escuela Polit´ecnica Superior de Algeciras Universidad de C´ adiz Av. Ram´ on Puyol s/n, 11202 Algeciras, Spain
Abstract Given a connected graph G = (V, E), a set of vertices S ⊂ V is an edge metric generator for G, if any two edges of G are identified by S by mean of distances to the vertices in S. Moreover, in a natural way, S is a mixed metric generator, if any two elements of G (vertices or edges) are identified by S by mean of distances. In this work we study the (edge and mixed) metric dimension of graphs. Keywords: mixed metric dimension, edge metric dimension, metric dimension.
Parameters related to distances in graphs have attracted the attention of several researchers since several years, and recently, one of them has centered several investigations, namely, the metric dimension. A vertex v of a connected graph G distinguishes two vertices u, w if d(u, v) = d(w, v), where d(x, y) represents the length of a shortest x − y path in G. A subset of vertices S of G is a metric generator for G, if any pair of vertices of G is distinguished by at least one vertex of S. The minimum cardinality of any metric generator for G is the metric dimension of G. This concept was introduced by Slater in [5] in connection with some location problems in graphs. On the other hand, 1
Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2016.10.047 1571-0653/© 2016 Elsevier B.V. All rights reserved.
I.G. Yero / Electronic Notes in Discrete Mathematics 55 (2016) 191–194
192
the concept of metric dimension was independently introduced by Harary and Melter in [2]. One can now consider the following situation. A metric generator uniquely recognizes the vertices of a graph in order to look out how they “behave”. However, what does it happen if there are anomalous situations occurring in some connections (edges) between some vertices? Is it possible that metric generators would properly identify the edges in order to also see their behaving? The answer to this question is negative. In connection with this, the following concepts deserve to be considered. Given a connected graph G = (V, E), a vertex v ∈ V and an edge e = uw ∈ E, the distance between the vertex v and the edge e is defined as dG (e, v) = min{dG (u, v), dG (w, v)}. A vertex w ∈ V distinguishes two edges e1 , e2 ∈ E if dG (w, e1 ) = dG (w, e2 ). A set S ⊂ V is an edge metric generator for G if any two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is the edge metric dimension and is denoted by edim(G) [3]. Moreover, a kind of mixed version of these two parameters described above is of interest. That is, a vertex v of G distinguishes two elements (vertices or edges) x, y of G if dG (x, v) = dG (y, v). Now, a set S ⊂ V is a mixed metric generator if any two elements of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is the mixed metric dimension and is denoted by mdim(G) [4].
1
Results
As stated, there are several graphs in which no metric generator is also an edge metric generator. In this sense, one could think that probably any edge metric generator S is also a standard metric generator. Nevertheless, this is again further away from the reality, although there are several graph families in which such facts occur. In [3], among other results, some comparison between these two parameters above were discussed. In contrast with this, for the case of mixed metric dimension, it clearly follows that that any mixed metric generator is also a metric generator and an edge metric generator. In this sense, the following relationship immediately follows. For any graph G, mdim(G) ≥ max{dim(G), edim(G)}. From now on, we present several results concerning the (edge, mixed) metric dimension of graphs. First of all, we remark the next complexity result. Theorem 1.1 [3] Computing the edge metric dimension of graphs is NP-hard. The result above was proved by using a reduction from the 3-SAT problem.
I.G. Yero / Electronic Notes in Discrete Mathematics 55 (2016) 191–194
193
Now, for the mixed metric dimension, nothing similar is known yet, although it is relatively natural to think that computing the mixed metric dimension is NP-hard, since also a similar fact occurs for the standard metric dimension. Based on these results, we next present some bounds or closed formulae for the (edge, mixed) metric dimension of several families of graphs. Proposition 1.2 [3] (i) For any integer n ≥ 2, edim(Pn ) = 1, edim(Cn ) = 2 and edim(Kn ) = n − 1. Moreover, edim(G) = 1 if and only if G is a path Pn . (ii) For any complete bipartite graph Kr,t different from K1,1 , edim(Kr,t ) = r + t − 2. Proposition 1.3 [4] Let G be any graph of order n. Then (i) mdim(G) = 2 if and only if G is a path. (ii) If at least one of the next situations happens, then mdim(G) = n. • Every vertex of G is a true twin vertex or an extreme vertex. • There are at least two vertices of degree n − 1. Proposition 1.4 [4] (i) For any integer n ≥ 4, mdim(Cn ) = 3.
⎧ ⎨ r + t − 1, if r = 2 or t = 2, (ii) For any integers r, t ≥ 2, mdim(Kr,t ) = ⎩ r + t − 2, otherwise. A vertex of degree at least 3 in a tree T is a major vertex of T . Any leaf u of T is said to be a terminal vertex of a major vertex v of T if d(u, v) < d(u, w) for every other major vertex w of T . The terminal degree of a major vertex v is the number of terminal vertices of v. A major vertex v of T is an exterior major vertex of T if it has positive terminal degree. Let n1 (T ) denote the number of leaves of T , and let ex(T ) denote the number of exterior major vertices of T . Proposition 1.5 [3] If T is a tree which is not a path, then edim(T ) = n1 (T ) − ex(T ). Proposition 1.6 [4] For any tree T , mdim(T ) = n1 (T ). The Cartesian product of two graphs G and H is the graph G2H, such that V (G2H) = {(a, b) : a ∈ V (G), b ∈ V (H)} and two vertices (a, b) and (c, d) are adjacent in G2H if and only if, either (a = c and bd ∈ E(H)), or (b = d and ac ∈ E(G)).
I.G. Yero / Electronic Notes in Discrete Mathematics 55 (2016) 191–194
194
Proposition 1.7 [3] For any integers r ≥ t ≥ 2, edim(Pr 2Pt ) = 2. Proposition 1.8 [4] For any integers r ≥ t ≥ 2, mdim(Pr 2Pt ) = 3. The next result shows an example where the edge metric dimension is strictly less than the standard metric dimension, since edim(C4r 2C4t ) = 4. Proposition 1.9 [3] For any integers r, t, edim(C4r 2C4t ) = 3. We close our exposition with a mathematical programming model for computing the mixed metric dimension of a graph G. A similar model for the metric dimension is known [1]. Let G be a graph of order n and size m with vertex set V = {v1 , . . . , vn } and edge set E = {e1 , . . . , em }. We consider the matrix D = [dij ] of order (n + m) × n such that dij = dG (xi , xj ), xi ∈ V ∪ E and xj ∈ V . Now, given the variables yj ∈ {0, 1} with j ∈ {1, 2, . . . , n} we define the following function: F(y1 , y2 , . . . , yn ) = y1 + y2 + . . . + yn . Clearly, minimizing the function F subject to the following constraints n
|dji − dli |yi ≥ 1 for every 1 ≤ j < l ≤ n + m,
i=1
is equivalent to finding a mixed metric basis of G, since the solution for y1 , y2 , . . . , yn represents a set of values for which the function F achieves the minimum possible.
References [1] Chartrand, G., L. Eroh, M. A. Johnson, and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99–113. [2] Harary, F., and R. A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191–195. [3] Kelenc, A., N. Tratnik, and I. G. Yero, Uniquely identifying the edges of a graph: the edge metric dimension, Manuscript, (2015). [4] Kelenc, A., D. Kuziak, A. Taranenko, and I. G. Yero, On the mixed metric dimension of graphs, Manuscript, (2016). [5] Slater, P. J., Leaves of trees, Congr. Numer. 14 (1975) 549–559.
Electronic Notes in Discrete Mathematics 55 (2016) 191–194 www.elsevier.com/locate/endm
Vertices, edges, distances and metric dimension in graphs Ismael Gonz´alez Yero 1 Departamento de Matem´ aticas, Escuela Polit´ecnica Superior de Algeciras Universidad de C´ adiz Av. Ram´ on Puyol s/n, 11202 Algeciras, Spain
Abstract Given a connected graph G = (V, E), a set of vertices S ⊂ V is an edge metric generator for G, if any two edges of G are identified by S by mean of distances to the vertices in S. Moreover, in a natural way, S is a mixed metric generator, if any two elements of G (vertices or edges) are identified by S by mean of distances. In this work we study the (edge and mixed) metric dimension of graphs. Keywords: mixed metric dimension, edge metric dimension, metric dimension.
Parameters related to distances in graphs have attracted the attention of several researchers since several years, and recently, one of them has centered several investigations, namely, the metric dimension. A vertex v of a connected graph G distinguishes two vertices u, w if d(u, v) = d(w, v), where d(x, y) represents the length of a shortest x − y path in G. A subset of vertices S of G is a metric generator for G, if any pair of vertices of G is distinguished by at least one vertex of S. The minimum cardinality of any metric generator for G is the metric dimension of G. This concept was introduced by Slater in [5] in connection with some location problems in graphs. On the other hand, 1
Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2016.10.047 1571-0653/© 2016 Elsevier B.V. All rights reserved.
I.G. Yero / Electronic Notes in Discrete Mathematics 55 (2016) 191–194
192
the concept of metric dimension was independently introduced by Harary and Melter in [2]. One can now consider the following situation. A metric generator uniquely recognizes the vertices of a graph in order to look out how they “behave”. However, what does it happen if there are anomalous situations occurring in some connections (edges) between some vertices? Is it possible that metric generators would properly identify the edges in order to also see their behaving? The answer to this question is negative. In connection with this, the following concepts deserve to be considered. Given a connected graph G = (V, E), a vertex v ∈ V and an edge e = uw ∈ E, the distance between the vertex v and the edge e is defined as dG (e, v) = min{dG (u, v), dG (w, v)}. A vertex w ∈ V distinguishes two edges e1 , e2 ∈ E if dG (w, e1 ) = dG (w, e2 ). A set S ⊂ V is an edge metric generator for G if any two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is the edge metric dimension and is denoted by edim(G) [3]. Moreover, a kind of mixed version of these two parameters described above is of interest. That is, a vertex v of G distinguishes two elements (vertices or edges) x, y of G if dG (x, v) = dG (y, v). Now, a set S ⊂ V is a mixed metric generator if any two elements of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is the mixed metric dimension and is denoted by mdim(G) [4].
1
Results
As stated, there are several graphs in which no metric generator is also an edge metric generator. In this sense, one could think that probably any edge metric generator S is also a standard metric generator. Nevertheless, this is again further away from the reality, although there are several graph families in which such facts occur. In [3], among other results, some comparison between these two parameters above were discussed. In contrast with this, for the case of mixed metric dimension, it clearly follows that that any mixed metric generator is also a metric generator and an edge metric generator. In this sense, the following relationship immediately follows. For any graph G, mdim(G) ≥ max{dim(G), edim(G)}. From now on, we present several results concerning the (edge, mixed) metric dimension of graphs. First of all, we remark the next complexity result. Theorem 1.1 [3] Computing the edge metric dimension of graphs is NP-hard. The result above was proved by using a reduction from the 3-SAT problem.
I.G. Yero / Electronic Notes in Discrete Mathematics 55 (2016) 191–194
193
Now, for the mixed metric dimension, nothing similar is known yet, although it is relatively natural to think that computing the mixed metric dimension is NP-hard, since also a similar fact occurs for the standard metric dimension. Based on these results, we next present some bounds or closed formulae for the (edge, mixed) metric dimension of several families of graphs. Proposition 1.2 [3] (i) For any integer n ≥ 2, edim(Pn ) = 1, edim(Cn ) = 2 and edim(Kn ) = n − 1. Moreover, edim(G) = 1 if and only if G is a path Pn . (ii) For any complete bipartite graph Kr,t different from K1,1 , edim(Kr,t ) = r + t − 2. Proposition 1.3 [4] Let G be any graph of order n. Then (i) mdim(G) = 2 if and only if G is a path. (ii) If at least one of the next situations happens, then mdim(G) = n. • Every vertex of G is a true twin vertex or an extreme vertex. • There are at least two vertices of degree n − 1. Proposition 1.4 [4] (i) For any integer n ≥ 4, mdim(Cn ) = 3.
⎧ ⎨ r + t − 1, if r = 2 or t = 2, (ii) For any integers r, t ≥ 2, mdim(Kr,t ) = ⎩ r + t − 2, otherwise. A vertex of degree at least 3 in a tree T is a major vertex of T . Any leaf u of T is said to be a terminal vertex of a major vertex v of T if d(u, v) < d(u, w) for every other major vertex w of T . The terminal degree of a major vertex v is the number of terminal vertices of v. A major vertex v of T is an exterior major vertex of T if it has positive terminal degree. Let n1 (T ) denote the number of leaves of T , and let ex(T ) denote the number of exterior major vertices of T . Proposition 1.5 [3] If T is a tree which is not a path, then edim(T ) = n1 (T ) − ex(T ). Proposition 1.6 [4] For any tree T , mdim(T ) = n1 (T ). The Cartesian product of two graphs G and H is the graph G2H, such that V (G2H) = {(a, b) : a ∈ V (G), b ∈ V (H)} and two vertices (a, b) and (c, d) are adjacent in G2H if and only if, either (a = c and bd ∈ E(H)), or (b = d and ac ∈ E(G)).
I.G. Yero / Electronic Notes in Discrete Mathematics 55 (2016) 191–194
194
Proposition 1.7 [3] For any integers r ≥ t ≥ 2, edim(Pr 2Pt ) = 2. Proposition 1.8 [4] For any integers r ≥ t ≥ 2, mdim(Pr 2Pt ) = 3. The next result shows an example where the edge metric dimension is strictly less than the standard metric dimension, since edim(C4r 2C4t ) = 4. Proposition 1.9 [3] For any integers r, t, edim(C4r 2C4t ) = 3. We close our exposition with a mathematical programming model for computing the mixed metric dimension of a graph G. A similar model for the metric dimension is known [1]. Let G be a graph of order n and size m with vertex set V = {v1 , . . . , vn } and edge set E = {e1 , . . . , em }. We consider the matrix D = [dij ] of order (n + m) × n such that dij = dG (xi , xj ), xi ∈ V ∪ E and xj ∈ V . Now, given the variables yj ∈ {0, 1} with j ∈ {1, 2, . . . , n} we define the following function: F(y1 , y2 , . . . , yn ) = y1 + y2 + . . . + yn . Clearly, minimizing the function F subject to the following constraints n
|dji − dli |yi ≥ 1 for every 1 ≤ j < l ≤ n + m,
i=1
is equivalent to finding a mixed metric basis of G, since the solution for y1 , y2 , . . . , yn represents a set of values for which the function F achieves the minimum possible.
References [1] Chartrand, G., L. Eroh, M. A. Johnson, and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99–113. [2] Harary, F., and R. A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191–195. [3] Kelenc, A., N. Tratnik, and I. G. Yero, Uniquely identifying the edges of a graph: the edge metric dimension, Manuscript, (2015). [4] Kelenc, A., D. Kuziak, A. Taranenko, and I. G. Yero, On the mixed metric dimension of graphs, Manuscript, (2016). [5] Slater, P. J., Leaves of trees, Congr. Numer. 14 (1975) 549–559.