Strong edge geodetic problem in networks

Abstract Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.


Introduction
Covering problems are among the fundamental problems in graph theory, let us mention the vertex cover problem, the edge cover problem, and the clique cover problem. An important subclass of covering problems is formed by path coverings that include the edge covering problem, the geodesic covering problem, the induced path covering problem and the path covering problem. Of a particular importance are coverings with shortest paths (also known as geodesics), e.g. in the analysis of structural behavior of social networks. In particular, the optimal transport flow in social networks requires an intensive study of geodesics [2][3][4]. In this paper we introduce and study a related problem that we call strong edge geodetic problem. This problem is in part motivated by the following application to social transport networks.
Urban road network is modeled by a graph whose vertices are bus stops or junctions. The urban road network is patrolled and maintained by road inspectors, see Fig. 1 for an example of a network with road inspectors I1, I2, I3, and I4. A road patrolling scheme is prepared satisfying the following conditions: 1. A road segment is a geodesic in the road network. It is patrolled by a pair of road inspectors by stationing one inspector at each end. 2. One pair of road inspectors is not assigned to more than one road segment. However, one road inspector is assigned to patrol other road segments with other inspectors. We point out that, under the assumption that the distances in a network between bus stops/junctions are integral, we may without loss of generality assume the network to be a simple graph. Indeed, to obtain an equivalent graph from the network, each edge may be subdivided by an appropriate number of times. From this reason, we are restricting ourselves in this paper to simple graphs. An example of a patrolling scheme for the network of Fig. 1  By condition 2., the restriction is that one pair of inspectors is assigned at most one road segment. For example, there are two road segments of equal length between inspectors I2 and I4, however these two inspectors are assigned a single road segment in the patrolling scheme. The strong edge geodetic problem is to identify a minimum number of road inspectors to patrol the urban road network. We proceed as follows. In the next section we state definitions and notions needed in this paper, and formally introduce the strong edge geodetic problem as well as two closely related problems. Then, in Section 3, we prove that the strong edge geodetic problem is NP-complete. In the subsequent section we discuss upper bounds on the strong edge geodetic number and show that it can be bounded from above by the edge isometric path number. In Section 5, we observe that simplicial vertices are intimately related to the strong edge geodetic problem and deduce several consequences, in particular determine exactly the strong edge geodetic number of block graphs and silicate networks. In Section 6, we introduce non-geodesic edges and use them to prove another lower bound on the strong edge geodetic number. The bound is shown to be in particular exact on glued binary trees without randomization.

Preliminaries
Let x and y be vertices of a graph G. Then the interval I.GI x; y/ (or I.x; y/ for short if G is clear from the context) between x and y is the set of vertices u such that u lies of some shortest x; y-path. In addition, for S Â V .G/ the geodetic closure I.S / of S is I.x; y/ : S is called a geodetic set if I.S / D V .G/. The geodetic problem, introduced by Harary et al. [1], is to find a geodetic set of G of minimum cardinality; this graph invariant is denoted with g.G/. Since then the problem has attracted several researchers and has been studied from different perspectives [5][6][7][8][9][10]. If x and y are vertices of a graph G, then let I e .GI x; y/ denote the set of the edges e such that e lies on at least one shortest x; y-path. Again we will simply write I e .x; y/ when there is no danger of confusion. For a set S Â V .G/, the edge geodetic closure I e .S / is the set of edges defined as x; y/ : A set S is called an edge geodetic set if I e .S / D E.G/. The edge geodetic problem, introduced and studied by Santhakumaran et al. [11], is to find a minimum edge geodetic set of G. The size of such a set is denoted with g e .G/. Note that g.G/ Ä g e .G/ holds for any graph G. As far as we know, the complexity status of the edge geodetic problem is unknown for the general case. On the other hand there are a significant number of theoretical results of the edge geodetic problem [12][13][14]. We now formally introduce the strong edge geodetic problem. If G is a graph, then S Â V .G/ is called a strong edge geodetic set if to any pair x; y 2 S one can assign a shortest x; y-path P xy such that By definition, in a strong edge geodetic set S there are jS j 2 paths P xy that cover all the edges of G. The cardinality of a smallest strong edge geodetic set S will be called the strong edge geodetic number G and denoted by sg e .G/. We will also say that the smallest strong edge geodetic set S is a sg e .G/-set. The strong edge geodetic problem for G is to find a sg e .G/-set of G. We emphasize that the strong edge geodetic problem requires, not only to determine a set S Â V .G/, but also a list of specific geodesics, that is, precisely one geodesic between each pair of vertices from S .
The Cartesian product G 1 G k of graphs G 1 ; : : : ; G k has the vertex set V .G 1 / V .G k /, vertices .g 1 ; : : : ; g k / and .g 0 1 ; : : : ; g 0 k / being adjacent if they differ in exactly one position, say in the i th, and g i g 0 i is an edge of G i [15]. A vertex v of a graph G is simplicial if its neighborhood induces a clique. In other words, a vertex v is simplicial if only if v lies in exactly one maximal clique. Finally, for a positive integer n we will use the notation OEn D f1; : : : ; ng.

Strong edge geodetic problem is NP-complete
In this section we prove: Theorem 3.1. The strong edge geodetic problem is NP-complete.
If G is a graph, then a set S of its vertices is called a shortest path union cover if the shortest paths that start at the vertices of S cover all the edges of G. Here, we consider all the shortest paths that start at v for each v 2 S . The shortest path union cover problem is to find a shortest path union cover of minimum cardinality. Boothe et al. [16] proved that the shortest path union cover problem is NP-complete. Now we show that the strong edge geodetic problem is NP-complete by a reduction from the shortest path union cover problem.
Hence, for each vertex u of G, vertices fu 1 ; : : : ; u deg.u/ g are added to V 1 and vertices fu 0 1 ; : : : ; u The construction is illustrated in Fig. 2. We can imagine that the graph G 0 D .V 0 ; E 0 / is composed of three layers where the top layer is the graph G, the middle layer is induced by the vertex set V 1 , and the bottom layer corresponds to the vertex set V 2 which is an independent set. Fig. 2. Graphs G and G 0 .
In the above construction, we thus create vertices w 0 1 ; : : : ; w 0 deg.w/ in G 0 for every vertex w of G. It is important to know the reason for this construction. Here is the explanation. It is easy to verify that in the example from Fig. 2 the singleton fvg is a shortest path union cover of G. Between the vertices v and z, there are two shortest paths vxz and vyz which cover the edges of G. See Fig. 3(a). Between one pair of vertices, more than one shortest path is allowed in the shortest path union cover problem. But in the strong edge geodetic problem between one pair of vertices v and z 0 1 , two shortest paths vxzz 1 z 0 1 and vyzz 1 z 0 1 are not allowed. See Fig. 3(b). In order to avoid this conflict, for every vertex w of G, we create w 0 1 ; : : : ; w 0 deg.w/ in G 0 . The shortest paths vxzz 1 z 0 1 and vyzz 2 z 0 2 that cover the edges of G 0 do not violate the condition of the strong edge geodetic problem. See Fig. 3(c). Here is a simple observation on the graph G 0 .V 0 ; E 0 /.
Then the edge u i u 0 i is not covered by any shortest path generated by the vertices of Y . Proof. Assuming that X is a shortest path union cover of G, we will prove that X [ V 2 is a strong edge geodetic set of G 0 . First we cover an edge Note that this also holds true in the case when u D v. The described paths in addition cover all the edges of E 3 too. Since X is a shortest path union cover of G, the edges of E are covered by the shortest paths uP v where u 2 X , v 2 V and P is a shortest path in G. Thus, the edges of E and E 1 are covered by the shortest paths uP vv Conversely, suppose that X [ V 2 is a strong edge geodetic set of G 0 . Then, for each edge e of E, there exists a shortest path P xy where x; y 2 X [ V 2 such that P xy covers e. By the structure of graph G 0 , for any shortest path P uv , if both u and v are not in V , then P uv will not cover any edge of E. Thus either x or y is in V . Say x 2 V . Thus a sub path of P xy starting at x which is also a shortest path indeed covers e. Hence, X is a shortest path union cover of G.
By Property 3.3 and 3.4, we have thus proved that a minimum shortest path union cover of G can be determined by finding a strong edge geodetic set of G 0 . Since G 0 can clearly be constructed from G in polynomial time, the argument is complete.

Upper bounds of sg e .G /
In this section an upper bound on sg e .G/ is given and another possible upper bound discussed. The upper bound given is in terms of (edge) isometric path covers, where an isometric path has the same meaning as a geodesic (alias a shortest path). Since the term isometric path cover is well-established, we use this terminology here.
Let G D .V; E/ be a graph. A set S of isometric paths of G is said to be an isometric path cover of G if every vertex v 2 V belongs to at least one path from S . The cardinality of a minimum isometric path cover is called the isometric path number of G and denoted by ip.G/ [17,18]. We now introduce the edge version of the isometric path cover in the natural way. A set S of isometric paths of a graph G D .V; E/ is an edge isometric path cover of G if every edge e 2 E belongs to at least one path from S. The cardinality of a minimum edge isometric path cover is called the edge isometric path number and denoted by ip e .G/. With this concept we have the following bounds.  . The inequality sg e .G/ Ä 2 ip e .G/ follows from the fact that an edge isometric path cover of cardinality ip e .G/ contains at most 2 ip e .G/ end-vertices.
Remark 4.2. The upper bound of sg e .G/ Ä 2 ip e .G/ is sharp. For instance, for the star graphs K 1;2r , it is easy to verify that sg e .K 1;2r / D 2r D 2 ip e .K 1;2r /. Additional sharpness examples can be constructed using tree-like graphs.
To conclude the section we note that the bound sg e .G/ Ä jV .G/j diam.G/ C 1 which one would be tempted to conjecture does not hold. For this sake consider the graph G obtained from the 6-cycle on the vertices v 1 ; : : : ; v 6 with natural adjacencies, by adding the edges v 1 v 3 and v 4 v 6 . Then diam.G/ D 3 yet it can be verified that sg e .G/ D 5.

Lower bound using simplicial vertices
In view of Theorem 3.1 it is natural to derive upper and lower bounds for the strong edge geodetic number. In this section we use simplicial vertices for this purpose and derive some related consequences.
Clearly, every simplicial vertex u belongs to every geodetic set, to every edge geodetic set, as well as to every strong edge geodetic set because u cannot be an inner vertex of a geodesic. We have already observed that g.G/ Ä g e .G/ holds for any graph G. In addition, we also have g e .G/ Ä sg e .G/. In the next result we collect these observations for the latter use.
Lemma 5.1. Let X be the set of simplicial vertices of a graph G. Then jXj Ä g.G/ Ä g e .G/ Ä sg e .G/ : Using Lemma 5.1, we compute strong edge geodetic number for certain graphs. First we proceed with block graphs and trees. Recall that a graph G is a block graph if every block of G is a clique.
Proposition 5.2. The set of simplicial vertices of a block graph G on at least two vertices is a sg e .G/-set of G.
Proof. Let X be the set of simplicial vertices of a block graph G. By Lemma 5.1, sg e .G/ jX j.
It remains to prove that X is a strong edge geodetic set of G. We proceed by induction on the number b of blocks of G. If b D 1 then G is a complete graph on at least two vertices, and the assertion clearly holds. Suppose now that b 2. Let Q be a pendant block of G and let V .Q/ D fx 1 ; : : : ; x k g, k 2. We may without loss of generality assume that x k is the cut vertex of Q. Then x i , i 2 OEk 1, are simplicial vertices of G. The graph G 0 induced by the vertices .V .G/ n V .Q// [ fx k g is a block graph (on at least two vertices) with one block less than G, hence by the induction hypothesis the set of its simplicial vertices forms a sg e .G/-set of G 0 . Let Y 0 be the set of geodesics that correspond to a sg e .G 0 /-set. Suppose first that x k is a simplicial vertex of G 0 . Then the set of paths forms a required strong edge geodetic set of G. Similarly, if x k is not a simplicial vertex of G 0 , then we replace every shortest path from Y 0 that is of the form P x k R with shortest paths P x k x i and x i x k R for all i 2 OEk 1 to find a required strong edge geodetic set of G also in this case.

Proposition 5.2 immediately implies:
Corollary 5.3. The set of leaves of a tree T is a unique sg e .G/-set of T .
Next we determine the strong edge geodetic number of hexagonal silicate networks which are well-known chemical networks. A hexagonal silicate network [19] is shown in Fig. 4. Proof. The simplicial vertices of G are marked by white bullets in Fig. 4. It is easy to verify that the set of simplicial vertices forms a strong edge geodetic set of G. The result follows from Lemma 5.1.
Corollary 5.4 considers only silicate networks of hexagonal type. This result can be extended to any type of silicate sheets. The verification is left to the reader.

Lower bound using convex components
For a different kind of a lower bound we introduce the following concept. We say that edges e and f of a graph G form a geodesic pair if they belong to some shortest path of G. Otherwise, e and f form a non-geodesic pair. Since (non)-geodesic pairs play a key role in Theorem 6.2, we next characterize such edges in the following result that might be of independent interest.

Proof.
Suppose that e and f are geodesic edges and let P be a shortest path containing these two edges. We may without loss of generality assume that d.v; x/ D d.u; y/ C 2 and set d.v; x/ D k. Then d.u; x/ D d.v; y/ D k C 1, so that fd.u; x/; d.u; y/; d.v; x/; d.v; y/g D fk; k C 1; k C 2g : Conversely, suppose that jfd.u; x/; d.u; y/; d.v; x/; d.v; y/gj D 3. We may without loss of generality assume that k D d.u; x/ is the minimal among the four distances. If also d.v; x/ D k, then d.u; y/ Ä k C 1 and d.v; y/ Ä k C 1 which is not possible. Therefore, d.v; x/ D k C 1. Since d.u; y/ Ä k C 1 it follows that d.v; y/ D k C 2. It now readily follows that e and f are geodesic edges.
We recall that a subgraph H of a graph G is convex, if for any vertices x; y 2 V .H /, every shortest x; y-path in G lies completely in H . Proof. Let GOEX 1 ; : : : ; GOEX t be the convex components of G F . Let U be an arbitrary strong edge geodetic set of G and set As GOEX i is convex, no shortest path between two vertices of U i contains an edge from F . Since in addition a shortest path between a vertex from U i and a vertex from U j , i ¤ j , contains at most one edge from F , if follows that X i ¤j jU i jjU j j jF j: (1) Using the fact that for any non-negative real numbers their arithmetic mean is at least as large as their geometric mean, we get Since jU k j 1 for k 2 OEt , the following inequality is straightforward: jU k j jU i jjU j j; i; j 2 OEt : Applying Inequality (3) for all t 2 pairs fi; j g we get Using (2), (4), and (1) in that order, we can now estimate as follows: Since U is an arbitrary strong edge geodetic set we conclude that Since sg e .G/ is an integral, the result follows.
To show that the bound of Theorem 6.2 is sharp, let us say that a graph G is r-good if it contains vertices u and v such that I.u; v/ contains r shortest u; v-paths which cover all the edges of G. For example, uniform theta graphs [20] are r-good graphs. To specify the vertices u and v we will denote such a graph with G v u . Clearly, an r-good graph G v u is necessarily bipartite and u and v are diametrical vertices of G. Proposition 6.3. Let G v i u i , i 2 OEn, be .n 1/-good graphs, n 2. Let V .K n / D OEn and let X be the graph obtained from the disjoint union of K n and G v i u i , i 2 OEn, by connecting u i with i for i 2 OEn. Then sg e .X / D n.
Proof. Clearly, a strong edge geodetic set of X must contain at least one vertex from each of the subgraphs G v i u i , hence sg e .X/ n.
To prove the other inequality it suffices to show that fv i W i 2 OEng is a sg e .G/-set of X . Let P i j , j 2 OEn n i , be the shortest v i ; u i -paths in G v i u i that cover all the edges. (Such paths exist because G v i u i is an .n 1/-good graph.) For any i Ä j let P ij be the path in X that is a concatenation of the paths P i j and P j i , and the edges i u i , ij , and j u j . It is straightforward to verify that each P ij is a shortest v i ; v j -path in G and that the paths P ij , i; j 2 OEn, i ¤ j , cover all the edges of X . Hence sg e .X / Ä n.
Consider the graph X of Proposition 6.3 and let F D E.K n /. Then F is a set of pairwise non-geodesic edges and X F consists of n convex components. Since jF j D n 2 , it follows that the bound of Theorem 6.2 is sharp for X . An important special case of Theorem 6.2 is the following. A set F of edges of a connected graph G is a convex edge-cut if G F consists of two convex components. Note that the edges of a convex edge-cut are pairwise non-geodesic for otherwise at least one of the components of G F would not be convex. Hence Theorem 6.2 immediately implies: Corollary 6.4. If F is a convex edge-cut of a graph G, then sg e .G/ l 2 p jF j m .
We next give an application of Corollary 6.4. Glued binary trees were introduced by physicists as a tool to design quantum algorithms [21] and quantum circuits [22]. It plays a significant role in Quantum Information Theory [23].
It is also used to study the transmission properties of continuous time quantum walks in quantum physics [24,25]. Lockhart et al. [26] designed glued tree algorithm using glued binary trees. An r-level complete binary tree T .r/ has 2 r leaves. An r-level glued binary tree GT .r/ is formed by connecting the leaves of two r-level complete binary trees T 1 .r/ and T 2 .r/. Fig. 5 displays two 3-level glued binary trees GT .3/. In general, the vertex set of GT .r/ is the union of the vertex sets of complete binary trees T 1 .r/ and T 2 .r/. Let L 1 and L 2 denote the vertex sets of leaves of T 1 .r/ and T 2 .r/ respectively. Notice that each set L 1 and L 2 is an independent set. The sets L 1 and L 2 induce a bipartite graph in GT .r/. Feder [23] classifies glued binary trees into those without randomization and those with randomization. If the edges of the bipartite subgraph induced by the sets L 1 and L 2 are in some fixed order, then they are called the glued binary trees without randomization, while if the sets L 1 and L 2 induce an arbitrary bipartite graph, then they are called the glued binary trees with randomization. Fig. 5(a) shows a 3-level glued binary tree without randomization. Fig. 5(b) displays a 3-level glued binary tree with randomization. Using Corollary 6.4, we can solve the strong edge geodetic problem for certain classes of glued binary trees without randomization. We define two graphs GT p .r/ and GT c .r/ which are glued binary trees without randomization. The graph GT p .r/ is obtained from T 1 .r/ and T 2 .r/ by adding straight edges between the corresponding leaves, see Fig. 6(a) for GT p .4/. The graph GT c .r/ is obtained from GT p .r/ by adding additional cross edges between the leaves as shown in Fig. 5(a) for the case GT c .3/. Theorem 6.5. sg e .GT p .r// D l 2 p 2 r m .

Conclusion
By modeling the urban road network problem as a graph combinatorial problem we proved that the urban road network problem is NP-complete. Naming this problem as the strong edge geodetic problem, we have studied the properties and characteristics of the strong edge geodetic problem from the perspectives of graph theory. We have derived some sharp lower bounds for strong edge geodetic number. Using these lower bounds, we have computed the strong edge geodetic number of certain classes of graphs such as trees, block graphs, silicate networks and some glued binary trees without randomization. The complexity of the problem is unknown for other graphs such as gridlike architectures, intersection graphs, Cayley graphs, chordal graphs and some of their subclasses (interval graphs, split graphs, k-tree, : : :), bipartite graphs and planar graphs.