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Strong edge geodetic problem in networks

  • Paul Manuel EMAIL logo , Sandi Klavžar , Antony Xavier , Andrew Arokiaraj and Elizabeth Thomas
Published/Copyright: October 3, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

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.

Keywords: Geodetic problem; Strong edge geodetic problem; Computational complexity; Transport networks
MSC 2010: 05C12; 05C70

1 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 [2–4]. 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.

Fig. 1 Urban road network patrolled by road inspectors I1, I2, I3, and I4.
Fig. 1

Urban road network patrolled by 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 is given below:

I1 and I2 patrol road segment 1−2−3−4−5−6−7;I1 and I3 patrol road segment 1−2−3−4−8−9−10−21−20−19;I1 and I4 patrol road segment 1−2−3−4−11−12−13;I2 and I3 patrol road segment 7−10−21−20−19;I2 and I4 patrol road segment 7−10−17−16−15−14−13; andI3 and I4 patrol road segment 19−18−15−14−13.

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.

2 Preliminaries

Let x and y be vertices of a graph G. Then the interval I(G; 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 ⊆ V(G) the geodetic closure I(S) of S is

I(S)=⋃{x,y}∈(s2)I(x,y).

S is called a geodetic set if I(S) = 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 [5–10].

If x and y are vertices of a graph G, then let Ie(G;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 Ie(x, y) when there is no danger of confusion. For a set S ⊆ V(G), the edge geodetic closure Ie(S) is the set of edges defined as

Ie(S)=⋃{x,y}∈(s2)Ie(x,y).

A set S is called an edge geodetic set if Ie(S) = 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 ge(G). Note that g(G) ≤ ge(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 [12–14].

We now formally introduce the strong edge geodetic problem. If G is a graph, then S ⊆ V(G) is called a strong edge geodetic set if to any pair x, y ∈ S one can assign a shortest x, y-path Pxy such that

⋃{x,y}∈(s2)E(Pxy)=E(G).

By definition, in a strong edge geodetic set S there are (|S|2) paths Pxy 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 sge(G). We will also say that the smallest strong edge geodetic set S is a sge(G)-set. The strong edge geodetic problem for G is to find a sge(G)-set of G. We emphasize that the strong edge geodetic problem requires, not only to determine a set S ⊆ V(G), but also a list of specific geodesics, that is, precisely one geodesic between each pair of vertices from S.

The Cartesian product G1 □ ⋯ □ Gk of graphs G1, …, Gk has the vertex set V(G1) × ⋯ × V(Gk), vertices (g1, …, gk) and (g1′,…,gk′) being adjacent if they differ in exactly one position, say in the ith, and gigi′ is an edge of Gi [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 [n] = {1, …, n}.

3 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 ∈ 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.

Given a graph G = (V, E), we construct a graph G′ = (V′, E′) with the vertex set V′ = V ∪ V1 ∪ V2 and the edge set E′ = E ∪ E1 ∪ E2 ∪ E3. The vertex set V1 is {ui : u ∈ V, i ∈[deg(u)]} and V2 is { ui′ : u ∈ V, i ∈ [deg(u)]}. Hence, for each vertex u of G, vertices {u1, …, udeg(u)} are added to V1 and vertices {u1′,...,udeg(u)′} are added to V2. The edge set E1 is {uui : u ∈ V, i ∈ [deg(u)]}. The edge set E2 is {uivj : u ∈ V, i ∈ [deg(u)] and v ∈ V, j ∈ [deg(v)]} ∪ {uiuj:u ∈ V, i, j ∈ [deg(u)]}. The edge set E3 is {ui ui′ :u ∈ V, i ∈ [deg(u)], i ≠ j}. The construction is illustrated in Fig. 2. We can imagine that the graph G′ = (V′, E′) is composed of three layers where the top layer is the graph G, the middle layer is induced by the vertex set V1, and the bottom layer corresponds to the vertex set V2 which is an independent set.

Fig. 2 Graphs G and G′.
Fig. 2

Graphs G and G′.

In the above construction, we thus create vertices w1′,...,wdeg(w)′ in G′ 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 {v} 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 z1′ , two shortest paths vxzz1 z1′ and vyzz1 z1′ are not allowed. See Fig. 3(b). In order to avoid this conflict, for every vertex w of G, we create w1′,...,wdeg(w)′ in G′. The shortest paths vxzz1 z1′ and vyzz2 z2′ that cover the edges of G′ do not violate the condition of the strong edge geodetic problem. See Fig. 3(c).

Fig. 3 The reason for constructing w1′,...,wdeg(w)′ $\begin{array}{} w_{1}', . . ., w_{{\text{deg}(w)}}' \end{array} $ in G′ for every vertex w of G.
Fig. 3

The reason for constructing w1′,...,wdeg(w)′ in G′ for every vertex w of G.

Here is a simple observation on the graph G′(V′, E′).

Observation 3.2

The vertex set V2 is a subset of any strong edge geodetic set Y of G′.

Proof

Suppose a vertex ui′ of V2 is not in Y. Then the edge ui ui′ is not covered by any shortest path generated by the vertices of Y. □

Property 3.3

For each strong edge geodetic set Y of G′, there exists a strong edge geodetic set Y′ of G′ such that Y′ ⊆ Y and Y′ = X ∪ V2 where X is a subset of V.

Proof

Suppose that Y contains a member ui of V1. By Observation 3.2, its corresponding vertex ui′ of V2 is in Y. Then the vertex ui can be replaced by ui′ which is already in Y. Set Y′ = Y ∖ V1. A shortest path uiP is a subpath of the shortest path ui′ ujP. Hence Y′ is is also a strong edge geodetic set of G′ and Y′ = X ∪ V2, where X is a subset of V. □

Property 3.4

X is a shortest path union cover of G if and only if X ∪ V2 is a strong edge geodetic set of G′.

Proof

Assuming that X is a shortest path union cover of G, we will prove that X ∪ V2 is a strong edge geodetic set of G′. First we cover an edge uivj of E2 by the shortest path ui′ uivj vj′ where ui′ and vj′ are in X ∪ V2. Note that this also holds true in the case when u = v. The described paths in addition cover all the edges of E3 too. Since X is a shortest path union cover of G, the edges of E are covered by the shortest paths uPv where u ∈ X, v ∈ V and P is a shortest path in G. Thus, the edges of E and E1 are covered by the shortest paths uPvvi vi′ where uPvvi vi′ is a shortest path generated by X ∪ V2. Thus X ∪ V2 is a strong edge geodetic set of G′.

Conversely, suppose that X ∪ V2 is a strong edge geodetic set of G′. Then, for each edge e of E, there exists a shortest path Pxy where x, y ∈ X ∪ V2 such that Pxy covers e. By the structure of graph G′, for any shortest path Puv, if both u and v are not in V, then Puv will not cover any edge of E. Thus either x or y is in V. Say x ∈ V. Thus a sub path of Pxy 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′. Since G′ can clearly be constructed from G in polynomial time, the argument is complete.

4 Upper bounds of sge(G)

In this section an upper bound on sge(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 = (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 ∈ 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 = (V, E) is an edge isometric path cover of G if every edge e ∈ 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 ipe(G). With this concept we have the following bounds.

Theorem 4.1

If G is a connected graph, then

1+(8×ipe(G)+1)2≤sge(G)≤2×ipe(G).

Proof

A strong edge geodetic cover S generates (|S|2) number of geodesics. Thus (|S|2) ≤ ipe(G) which in turn implies that |S|≥1+(8×ipe(G)+1)2. Since this inequality is true for every strong edge geodetic cover, we conclude that sge(G)≥1+(8×ipe(G)+1)2.

The inequality sge(G) ≤ 2 × ipe(G) follows from the fact that an edge isometric path cover of cardinality ipe(G) contains at most 2 × ipe(G) end-vertices. □

Remark 4.2

The upper bound of sge(G) ≤ 2 × ipe(G) is sharp. For instance, for the star graphs K1,2r, it is easy to veriffi that sge(K1,2r) = 2r = 2 × ipe(K1,2r). Additional sharpness examples can be constructed using tree-like graphs.

To conclude the section we note that the bound sge(G) ≤ |V(G)|–diam(G)+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 v1, …, v6 with natural adjacencies, by adding the edges v1 V3 and v4v6. Then diam (G) = 3 yet it can be verified that sge(G) = 5.

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) ≤ ge(G) holds for any graph G. In addition, we also have ge(G) ≤ sge(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

|X|≤g(G)≤ge(G)≤sge(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 sge(G)-set of G.

Proof

Let X be the set of simplicial vertices of a block graph G. By Lemma 5.1, sge(G) ≤ |X|.

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 = 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) = {x1, …, xk}, k ≤ 2. We may without loss of generality assume that xk is the cut vertex of Q. Then xi, i ∈ [k−1], are simplicial vertices of G. The graph G′ induced by the vertices (V(G) ∖ V(Q)) ∪ {xk} 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 sge(G)-set of G′. Let Y′ be the set of geodesics that correspond to a sge(G′)-set. Suppose first that xk is a simplicial vertex of G′. Then the set of paths {Pxi : P ∈ Y′,P = xkP′,i ∈ [k−1]} ∪ { P : P ∈ Y′ ,xk is not end vertex of P} ∪ {xixj : i,j ∈ [k−1],i ≠ j} forms a required strong edge geodetic set of G. Similarly, if xk is not a simplicial vertex of G′, then we replace every shortest path from Y′ that is of the form PxkR with shortest paths Pxkxi and xixkR for all i ∈ [k−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 sge(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.

Fig. 4 Silicate network.
Fig. 4

Silicate network.

Corollary 5.4

The simplicial vertices of the hexagonal silicate network G form a sge(G)-set of G.

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.

6 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.

Proposition 6.1

Let e = uv and f = xy be edges of a connected graph G. Then e and f are geodesic if and only if|{d(u, x),d(u,y),d(v,x),d(v,y}| = 3.

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(u,y)+2 and set d(v,x) = k. Then d(u,x) = d(v,y) = k+1, so that

{d(u,x),d(u,y),d(v,x),d(v,y)}={k,k+1,k+2}.

Conversely, suppose that |{d(u, x),d(u,y),d(v,x),d(v,y}| = 3. We may without loss of generality assume that k = d(u,x) is the minimal among the four distances. If also d(v,x) = k, then d(u,y) ≤ k+1 and d(v,y) ≤ k+1 which is not possible. Therefore, d(v,x) = k+1. Since d(u,y) ≤ k+1 it follows that d(v,y) = k+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 ∈ V(H), every shortest x, y-path in G lies completely in H.

Theorem 6.2

Let F be a set of pairwise non-geodesic edges of a graph G. If G– F consists of t ≥ 2 convex components, then

sge(G)≥⌈t×2|F|t(t−1)t⌉.

Proof

Let G[X1], …, G[Xt] be the convex components of G – F. Let U be an arbitrary strong edge geodetic set of G and set Ui = U ∩ Xi, i ∈ [t]. Clearly, U=∪i=1tUi and Ui ∩ Uj = ∅ for i ≠ j. As G[Xi] is convex, no shortest path between two vertices of Ui contains an edge from F. Since in addition a shortest path between a vertex from Ui and a vertex from Uj, i ≠ j, contains at most one edge from F, if follows that

∑i≠j|Ui||Uj|≥|F|. (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

|U|=∑k=1t|Uk|≥t×∏k=1t|Uk|t. (2)

Since |Uk| ≤ 1 for k ∈ [t], the following inequality is straightforward:

∏k=1t|Uk|≥|Ui||Uj|,i,j∈[t]. (3)

Applying Inequality (3) for all t2 pairs {i, j} we get

t2∏k=1t|Uk|≥∑i≠j|Ui||Uj|. (4)

Using (2), (4), and (1) in that order, we can now estimate as follows:

|U|≥t×∏k=1t|Uk|t≥t×1(2t)∑i≠j|Ui||Uj|t≥t×1(2t)|F|t.

Since U is an arbitrary strong edge geodetic set we conclude that

sge(G)≥t×2|F|t(t−1)t.

Since sge(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 Guv . Clearly, an r-good graph Guv is necessarily bipartite and u and v are diametrical vertices of G.

Proposition 6.3

Let Guivi , i ∈ [n], be (n−1)-good graphs, n ≥ 2. Let V(Kn) = [n] and let X be the graph obtained from the disjoint union of Kn and Guivi , i ∈ [n], by connecting ui with i for i ∈ [n]. Then sge(X) = n.

Proof

Clearly, a strong edge geodetic set of X must contain at least one vertex from each of the subgraphs Guivi , hence sge(X) ≤ n.

To prove the other inequality it suffices to show that {vi : i ∈ [n]} is a sge(G)-set of X. Let Pji , j ∈ [n] ∖ i, be the shortest vi, ui -paths in Guivi that cover all the edges. (Such paths exist because Guivi . is an (n−1)-good graph.) For any i ≤ j let Pij be the path in X that is a concatenation of the paths Pji and Pij , and the edges iui, ij, and juj. It is straightforward to verify that each Pij is a shortest vi, vj -path in G and that the paths Pij, i,j ∈ [n], i ≠ j, cover all the edges of X. Hence sge(X) ≤ n. □

Consider the graph X of Proposition 6.3 and let F = E(Kn). Then F is a set of pairwise non-geodesic edges and X – F consists of n convex components. Since |F| = n2 , 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 sge(G)≥⌈2|F|⌉.

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 2r leaves. An r-level glued binary tree GT(r) is formed by connecting the leaves of two r-level complete binary trees T1(r) and T2(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 T1(r) and T2(r). Let L1 and L2 denote the vertex sets of leaves of T1(r) and T2(r) respectively. Notice that each set L1 and L2 is an independent set. The sets L1 and L2 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 L1 and L2 are in some fixed order, then they are called the glued binary trees without randomization, while if the sets L1 and L2 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.

Fig. 5 (a) Glued binary trees without randomization GTc(r) and (b) glued binary trees with randomization.
Fig. 5

(a) Glued binary trees without randomization GTc(r) and (b) glued binary trees 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 GTp(r) and GTc(r) which are glued binary trees without randomization. The graph GTp(r) is obtained from T1(r) and T2(r) by adding straight edges between the corresponding leaves, see Fig. 6(a) for GTp(4). The graph GTc(r) is obtained from GTp(r) by adding additional cross edges between the leaves as shown in Fig. 5(a) for the case GTc(3).

Fig. 6 Glued binary tree GTp(4).
Fig. 6

Glued binary tree GTp(4).

Theorem 6.5

sge(GTp(r)) = ⌈22r⌉ .

Proof

The set of straight edges (dotted edges in Fig. 6(b)) between the leaves forms a convex edge-cut of cardinality 2r. By Corollary 6.4, sge(GTp(r)) ≤ ⌈22r⌉ .

Let k = ⌈22r⌉ .The vertex set of GTp(r) is the union of the vertex sets of complete binary trees T(r) and T2(r). Starting from the root of T1(r), select k/2 vertices of T1(r) in a breadth first search (BFS) traversal order. Name this set as S1. In the same way, build a set S2 by selecting another k/2 vertices from T2(r). Set S = S1 ∪ S2. The red bullets in Fig. 6(b) represent this set S. It is a simple exercise to verify that the set S of vertices forms a strong edge geodetic set. Thus sge(GTp(r)) = ⌈22r⌉ . □

Using an approach from the proof of Theorem 6.5 we can also deduce the following result.

Theorem 6.6

sge(GTc(r))=⌈22×2r⌉.

Further research is to identify the classes of glued binary trees without randomization for which the strong edge geodetic problem can be solved. The strong edge geodetic problem for glued binary trees with randomization is left as an open problem.

7 Further research

The vertex version of the strong edge geodetic problem is the strong geodetic problem defined analogously as follows. A set S ⊆ V(G) is a strong geodetic set if for any pair x, y ∈ S there exists a shortest x, y-path Pxy such that

⋃x,y∈(s2)V(Pxy)=V(G).

The strong geodetic problem is to find a smallest strong geodetic set of G. To our knowledge, there is no literature yet on the strong geodetic problem. We believe it is also NP-complete, similar to the edge version of the problem. As already mentioned in Section 4, Fitzpatrick et al. [17] have studied the isometric path problem. It would be interesting to investigate relationships between the strong geodetic problem and the isometric path problem. Fitzpatrick et al. [17] have derived the isometric path number for two dimensional grids. Further research would be to design techniques to derive the strong geodetic number for grid-like architectures and similar architectures.

8 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.

Acknowledgements

This work was supported and funded by Kuwait University, Research Project No. (QI 01/16). We thank Xuding Zhu for suggesting the term “strong edge geodetic problem”.

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Received: 2017-2-15
Accepted: 2017-8-16
Published Online: 2017-10-3

© 2017 Manuel et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0101
Keywords for this article
Geodetic problem; Strong edge geodetic problem; Computational complexity; Transport networks
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BY-NC-ND 3.0

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