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6URAL MATHEMATICAL JOURNAL, Vol. 8, No. 1, 2022, pp. 64-75
DOI: 10.15826/umj.2022.1.007
ON DOUBLE SIGNAL NUMBER OF A GRAPH
X. Lenin Xaviour
Department of Mathematics, Nesamony Memorial Christian College, Marthandam - 629165, Tamil Nadu, India [email protected]
S. Ancy Mary
Department of Mathematics, St. John's College of Arts and Science, Ammandivilai,
Affliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627012, Tamil Nadu, India [email protected]
Abstract: A set S of vertices in a connected graph G = (V, E) is called a signal set if every vertex not in S lies on a signal path between two vertices from S. A set S is called a double signal set of G if S if for each pair of vertices x, y € G there exist u, v € S such that x, y € L[u, v]. The double signal number dsn (G) of G is the minimum cardinality of a double signal set. Any double signal set of cardinality dsn (G) is called dsn-set of G. In this paper we introduce and initiate some properties on double signal number of a graph. We have also given relation between geodetic number, signal number and double signal number for some classes of graphs.
Keywords: Signal set, Geodetic set, Double signal set, Double signal number.
1. Introduction
By a graph G = (V,E) we mean a finite, connected, undirected graph with neither loops nor multiple edges. The order |V| and size |E| of G are denoted by p and q respectively. For graph theoretic terminology we refer to [1]. The open neighborhood of any vertex v in G is N(v) = {x : xv € E(G)} and closed neighborhood of a vertex v in G is N[v] = N(v) U{v}. The degree of a vertex in the graph G is denoted by deg (v) and the maximum degree (minimum degree) in the graph G is denoted by A(G)(5(G)). For a set S C V(G) the open (closed) neighborhood N(S)(N[S]) in G is defined as
N(S) = U N(v)(N[S] = U N[v]).
v€S v€S
A graph G is said to be connected if any two vertices in G are joined by a path. A maximal connected subgraph of G is called a component of G. A graph is said to be disconnected if it has at least two components. A cut-vertex of a connected graph is a vertex whose removal results a disconnected graph. A graph G is said to be regular if every vertex of G has equal degree.
If G is a connected graph the distance d(x, y) is the length of a shortest x-y path in G. The diameter is defined by diam (G) = maxxy€V(G) d(x,y). Two vertices u and v are said to be antipodal vertices if d(u, v) = diam (G). If e = {u, v} is an edge of a graph G with deg (u) = 1 and deg (v) > 1, then we call e a pendant edge, u a pendant vertex and v a support vertex. A vertex v of G is said to be an extreme vertex if the subgraph induced by its neighborhood is complete. The set of all extreme vertices is denoted by Ext (G). An acyclic connected graph is called a tree. An x-y path of length d(x, y) is called geodesic.
A set S C V(G) is called a geodetic set of G, if every vertex in G lies on a geodesic joining a pair of vertices of S. The geodetic number of G, denoted by g(G), is the minimum cardinality of a geodetic set of G. The geodetic number of a disconnected graph is the sum of the geodetic number of its components. Any geodetic set of cardinality g(G) is called g-set of G.
A set S of vertices in G is called a double geodetic set of G if for each pair of vertices of G lie on any geodesic joining pair of vertices from S. The double geodetic number dg (G) is the minimum cardinality of a double geodetic set. Any geodetic set of cardinality dg (G) is called dg-set of G. The double geodetic number of a graph was introduced and studied in [7]. Various concepts inspired by geodetic sets are introduced in [1, 3, 4].
On a various study on the distance in graphs, we refer to [1]. In the meantime, Chartrand et al. introduced a new type of distance parameter called the detour distance in graphs. Once a new type of distance between two vertices was introduced by Chartrand et al., various new distance parameters such as Supreme distance, D-distance and many more, were introduced by different researchers. In continuation, Kathiresan et al. introduced a distance parameter, called the signal distance in graphs [5]. The signal distance dSD(u,v) between a pair of vertices u and v is defined
where S is a path connecting u and v, d(u,v) is the length of the path S and the sum ^weV(G), where sum runs over all the internal vertices between u and v in the path S. The u-v signal path of length dSD (u,v) is also called geosig. A vertex v is said to lie on a geosig P if v is an internal vertex of P. The signal interval L[x,y] consists of x,y and all vertices lying on some x-y geosig of G and for a non empty set S C V(G), L[S] = (Jx yeS L[x,y].
A set S C V(G) in a connected graph is a signal set of G if L[S] = V(G). The signal number sn (G) is the minimum cardinality of a signal set of G. A signal set of cardinality sn (G) is called a sn-set of G. The signal number of a graph was introduced in [8] and further studied in [2, 5]. The concept of signal number can be applied in the fields of electrical engineering and irrigation systems. It was shown that the determining the signal number of a graph is an NP-hard problem. Let 2V denote the set of all the subsets of V. The mapping L: V x V ^ 2V defined by
is the signal function of G. One of the basic properties of L is that x,y € L[x,y] for any pair x,y € V. Hence the signal function captures every pair of vertices and so the problem of double signal sets is trivially well-defined while it is clear that this fails in many graphs already for triplets (for example, complete graphs). This is the motivation for introducing and studying double signal sets.
The concepts of distance in graphs is a major component in graph theory with its convexity concepts having numerous applications in real life problems. There are several interesting applications of these concepts to facility location in real life situations, routing of transport problems and communication network designs. As the path involved in this discussion of this paper are geosig, no intervention by hackers or enemies is possible to the respective facilities provided. Further, as signal paths are secured and longer than geodesic paths, it is advantageous to more customers in getting the service with protection.
The following theorems will be used in the subsequent sections.
Theorem 1 [2]. For any connected graph G, the set of all end vertices is a subset of every signal set of G.
by
wev (G)
L[x, y] = {z G V : z lies on a x-y geosig in G}
Theorem 2 [3]. Each extreme vertex of a connected graph G belongs to every geodetic set of G.
Theorem 3 [7]. Each extreme vertex of a connected graph G belongs to every double geodetic set of G.
The signal number of some standard classes of graph can be easily found and are given below:
• Path Pp of p > 2 vertices, sn (Pp) = 2.
n i t ^o i2' if P iseven'
• Cycle Cp of p > 3 vertices, sn (Cp) = <
I 3' if p is odd.
• Complete graph Kp of p > 2 vertices, sn (Kp) = p.
• Peterson graph G, sn (G) = 4.
• Star graph Ki,p_i of p > 2 vertices, sn (Ki,p_i) = p — 1.
(mm if mm ^ 3 4, otherwise.
2. Double signal number of a graph
Definition 1. Let G be a connected graph with at least two vertices. A set S of vertices of G is called a double signal set of G if for each pair of vertices € G there exist u, v € S such that € L[u, v]. The double signal number dsn(G) of G is the minimum cardinality of a double signal set. Any double signal set of cardinality dsn(G) is called dsn-set of G.
Example 1. For the graph G in Fig. 1, it is clear that no 2-element subset of G is a signal set of G. Now S = (vi, v4, v5} is a signal set of G and so sn (G) = 3. Clearly the pair of vertices v3, v6 lies only the v3-v6 geosig. Similarly, the vertices v6,vs lies only the v6-vs geosig. Also the vertices v2, v6 and v6, v7 lies only the v2-v6 and v6, v7 geosig, respectively. Therefore that S is not a double signal set of G. Since v2 ,v3 ,v7, v8 be an internal vertices of v1 -v4 geosig path, we need at least 6
V2 V3
vertices to form a double signal set of G and so dsn(G) > 6. Now, since S1 = (v1, v3, v4, v5, v6, v8} is a double signal set, it follows that dsn (G) = 6.
Remark 1. For the graph G in Fig. 1, S = (v1,v4} is the unique g-set and dg-set of G and so g(G) = dg (G) = 2. Thus the double signal number is different from geodetic number and double geodetic number.
The following theorem directly follows by the definition of signal number and double signal number.
Theorem 4. For any connected graph G of order p, 2 < sn (G) < dsn (G) < p.
Remark 2. The bounds in Theorem 4 are sharp. For the complete graph Kp(p > 2), dsn(KP) = p. The set of the two end vertices of path graph Pp(p > 2) forms a unique double signal set and so dsn(PP) = 2. Thus the nontrivial complete graph Kp has the largest possible double signal number and the nontrivial path graph Pp has the smallest double signal number. Also Example 2 shows that the bounds in Theorem 4 is sharp.
Theorem 5. Each extreme vertex of a connected graph G belongs to every signal set of G.
Proof. Let u be an extreme vertex of G and let S be a signal set of G. If u is an end-vertex, then by Theorem 1 u € S. Suppose u / S be non end-vertex. Then u is an internal vertex of an x-y geosig path, say P, for some x,y € S. Since deg(u) > 2, u has at least two neighbours in P which are not adjacent and so that u is not an extreme vertex, which is a contradiction. Hence u € S. □
The following result is an easy consequences of Theorem 5.
Result 1. For the complete graph Kp(p > 2), dsn(KP) = p.
To aid in our discussion throughout this paper, we define a definition as follows.
Definition 2. Let G be a connected graph of order p > 2. A vertex v € G is said to be a weak extreme vertex, if there exists a vertex u in G such that v is either an initial vertex or a terminal vertex of any signal interval containing both u and v.
Theorem 6. Every double signal set of a connected graph G contains all the weak extreme vertices of G. In particular, if the set S of all weak extreme vertices is a double signal set, then S is the unique dsn-set of G.
Proof. Let S be a double signal set of G and let x be a weak extreme vertex of G. Suppose x / S. Let y be any vertex in G such that x = y. Since S is a double signal set of G, we have for some u,v € S, that x, y lie on an u-v geosig path. Also, that x is a weak extreme vertex of G shows either x = u or x = v. It follows that x € S, which is a contradiction. □
Corollary 1. Each extreme vertex of a connected graph G belongs to every double signal set of G.
Proof. Since every extreme vertex of G is weak extreme, the result follows from Theorem 6.
□
Example 2. For the graph G in Fig. 2, the set S = {v1,v5,v7} of extreme vertices form unique minimum signal set of G and so sn(G) = 3. Since the pair of vertices v3,v6 does not lie on any geosig of any pair of vertices from S, that S is not a double signal set of G. Also the vertex v6 is the only non-extreme vertex which became weak extreme. It is clear that the set S1 = S U {v6} of all weak extreme vertices form a double signal set of G and so by Theorem 6 dsn (G) = 4.
Result 2. For any cycle Cp(p > 3),
2, if p is even,
dsn(CP)= ., . ,,
\p, if p is odd.
Result Result Result
Result
Figure 2. Graph G.
3. For any wheel Wp = K + Cp_1(p > 3), dsn (Wp) = p.
4. For the complete bipartite graph Km,n(m,n > 2), dsn(Km,n) = min(m,n}.
5. For any fan Fp = K1 + Pp-1(p > 3),
. . |p — 2, if p is even,
dsn(FP) = S ., . ,,
I p, ii p is odd.
6. For the star graph K1,P-1, dsn (K1,P-1) = p — 1.
Theorem 7. Let G be a connected graph with cut vertices and let S be a double signal set of G. If v is a cut vertex of G, then every component of G — v contains at least one element of S.
Proof. Let v be cut vertex of G and S be a double signal set of G. Suppose to the contrary, there exists a component, say H of G — v such that H contains no vertex of S. By Theorem 6, S contains all the weak extreme vertices of G and hence, by assumption H does not contain any weak extreme vertex of G. Let u € V(H). Since, S is a double signal set of G, there exist vertices x, y € S such that u, v € L[x,y] C L[S]. Let the x-y geosig path in G be P : x = uo,u1, ..,u, = y such that u = x, y. Since, v is a cut vertex, the x-u subpath of P and the u-y subpath of P both contain v, it implies that P is not a geosig path, which is a contradiction. Hence, every component of G — v contains an element of S. □
Theorem 8. No cut-vertex of a connected graph G belongs to any dsn-set of G.
Proof. Suppose S be a dsn-set of a connected graph G that contains a cut-vertex v. Let G1, G2,..., Gn(n > 2) be the components of G — v. Let S1 = S — {v}. We show that S1 is a double signal set of G. Let x,y € V(G). Since S is a double signal set, then x,y lies on a geosig P joining a pair of vertices a, b € S. If v / (a, b}, then (a, b} C S1 and so that S1 is a double signal set of G, which contradicts the minimality of S. Therefore, assume that v € (a, b} such that v = b and a € G1. Since S1 C S, that a € S1. By Theorem 7 we can fix a vertex u € Gk for k = 1 such that u € S. Since u = v, that u € S. Now, since v is a cut vertex of G, the signal interval of the path between a and v contained in the signal interval of the path between a and u. This shows that x,y lies on the geosig between a, u € S1. Therefore, that S1 is a double signal set of G, which again contradicts the minimality of S. Hence no cut-vertex of G belongs to any dsn-set of G. □
Definition 3. Let u be a vertex in G. A vertex v in G is said to be an u — signal vertex if for any vertex w = u, v with dSD (u, v) < dSD (u, w), w lies on an u-v signal path.
Theorem 9. For any connected graph G, sn (G) = 2 if and only if there exist vertices u, v such that v is an u-signal vertex of G.
Proof. Let sn (G) = 2 and S = {u, v} be a sn-set of G. Then every vertex w in G lies on this u-v signal path and so that dSD(u,v) is minimum. Thus, dSD(u,v) < dSD(u,w) for every w = u,v. Hence v is an u-signal vertex of G. The converse part is obvious. □
Theorem 10. For a nontrivial connected graph G, dsn(G) = 2 if and only if sn(G) = 2.
Proof. Let S = {u,v} be a sn-set of G such that sn(G) = 2. Then every pair of vertices of G lies on a u-v geosig and so that S itself forms a double signal set. Hence, dsn(G) = 2. Converse part follows from Theorem 4. □
The following result follows from Theorem 6 and Theorem 8.
Result 7. If T is a tree with l end vertices, then dsn (T) = l. In fact, the set of all end vertices of T is the unique dsn-set of T.
Lemma 1. Let G be a connected graph of order p > 2. If there exists a vertex v € G such that
deg (v) > ^ deg (w) + l(P),
weG
where l(P) is the length of a geosig path P between any two antipodal vertices and the sum J2wec deg (w) runs over all the internal vertices between the antipodal vertices in P, then dsn (G) = p.
For every connected graph G, it is clear that rad(G) < diam(G) < 2 rad (G) [6]. Ostrand showed that any two positive integers a and b with a < b < 2a are realizable as the radius and diameter, respectively. This theorem can be extended so that the double geodetic number can be prescribed as well.
Theorem 11. For positive integers r,d and a > 2 with r < d < 2r, there exists a connected graph G with rad (G) = r, diam (G) = d and dsn (G) = a.
Proof. If r = 1, then consider G = Ka or G = K1,a according to whether d = 1 or d = 2, respectively. If r = d > 2 and a = 2, then we take G = C2r.
Now assume that r = d > 2 and a > 3. Let C2r : u1,u2, ...,ur,ur+1, ...,u2r,u1 be a cycle of order 2r. Add p — 1 pendant edges v1u1,v2u1, ...,va-1 u1 to obtained the graph G. Clearly rad (G) = diam(G) = r. The graph G has a — 1 extreme vertices, that is, S = {v1,v2, ...,va-1}. By Corollary 1, each double signal set of G must contain S and that L[S] = V(G). Hence, dsn (G) > a — 1. On the other hand, we have L[S U {ur+1}] = V(G) and every pair of vertices of G lies on a geosig of some pair of vertices from S U {ur+1}, implying that dsn (G) = a.
Finally assume 2 < r < d. First assume a > 3. Let G be the graph obtained from the disjoint union of a cycle C2r : u1,u2, ...,ur,ur+1 ,...,u2r,u1 of order 2r and a path Pd-r+1 : v0,v1, ...,vd-r of order d — r + 1 by identifying u1 and vo. Add new pendant edges urw1,urw2,urw3, ...,urwa-3. Then G has radius r and diameter d. This graph G is shown in Fig. 3.
Now, we prove the set {w1 ,w2, ....wa-3,ur+1,u2r,vd-r} forms a double signal set of G. By Corollary 1, w1 ,w2, ...,wa-3,vd-r € S, where S is a double signal set of G. Further, as vertices ur+1,ur+2, ...,u2r in V(G) — {w1,w2, ...,wa-3,vd-r} cannot covered by using the vertices w1 ,w2, ...,wa-2,vd-r, |S| > a — 3 + 1 = a — 2. Now it is clear that ur+1 is either an internal
Ur+2
ur+1
' -Wa-3
— v0 V\ v2 V3
Ud-r
Ur
U3 U2
Figure 3. Graph G.
vertex or a terminal vertex of any signal path containing the pair of vertices Similarly,
u2r is either an internal vertex or a terminal vertex of any signal path containing the pair of vertices u2r, Wj. Thus ur+1,u2r are weak extreme vertices. Therefore by Theorem 6, ur+1,u2r € S and so dsn(G) > a. Since {w1, w2, ....wa-3,ur+1,u2r,vd-r} forms a double signal set of G, that dsn(G) = a. For the case a = 2, we remove the pendant edges urw1,urw2,urw3, ...,urwa-3 of G in Fig. 3. Clearly G has radius r and diameter d. Also {ur+1,vd-r} is the unique double signal set of G and so by Theorem 4 we conclude that dsn(G) = 2. This complete the proof. □
Theorem 12. For every pair a,p of integers with 2 < a < p, there exists a connected graph G of order p such that dsn (G) = a.
Proof. If 2 < a = p, we take G = Kp. For 2 < a < p, we consider a tree graph G of order p with a end-vertices. □
3. The double signal number and double geodetic number of a graph
In this section, we consider the realization result connecting the double signal number and double geodetic number of connected graphs. For this, first we focus the signal number and geodetic number. Because, for the graph G in Fig. 2, g(G) = 3 and sn(G) = 3 so that sn(G) = g(G). Similarly, for the graph G in Fig. 1, g(G) = 2 and sn(G) = 3 so that sn(G) > g(G) and for the graph G in Fig. 4, g(G) = 3 and sn (G) = 2 so that sn (G) < g(G).
v2 v3
It is easily seen that a signal set is not in general a geodetic set in a graph G. Also the converse ir true. We verify that if S is a signal set and D is a geodetic set of G, either S C D or D C S. Hence the signal set and the geodetic set depend one to another. Therefore we can't find out which one is the bigger set.
Result 8. If G is a tree, then sn (G) = g(G).
Theorem 13. If G is a regular graph, then sn(G) = g(G).
Proof. Since G is regular, the degree of every vertices of G is unique. So the signal distance between any pair of vertices depends only the geodesic distance between this pair of vertices.
In view of Theorem 4, we have the following realization theorems.
Theorem 14. For any integers a, b and c with 3 < a < b < c, there exists a connected graph G with g(G) = a, sn (G) = b and dsn (G) = c.
Proof. This theorem is proved by considering four cases.
Case 1. a = b = c. Then for the complete graph Ka, g(G) = sn (G) = dsn (G) = a.
Case 2. a = b < c. let G be the graph in Fig. 5 obtained from the path P3 : u1,u2,u3 of order 3, by adding c vertices v1,v2, ...,va-2,w1,w2,..., wc-a+2 to P3 and joining each vertex vi (1 < i < a—2) to u2; and joining each vertex wi(1 < j < c — a + 2) to u1 and u3. By Theorem 2, Theorem 5 and Corollary 1, every geodetic set, every signal set and every double signal set of G contains the set S = {v1,v2, ...,va-2} of all extreme vertices of G. Clearly, S is not a geodetic set of G. Also, for any x € V(G) — S, S U {x} is not a geodetic set or a signal set of G and so g(G) > a. Now it is easy to check that S1 = S U {u1,u2} is a geodetic set of G. Since every vertex in V(G) — S1 lies on the signal path between some vertices from S1, so S1 is the minimum geodetic set as well as signal set of G. Thus g(G) = sn (G) = a. It is clear that the pair of vertices vi, wj for (1 < i < a — 2) and (1 < j < c — a + 2) do not lie on any u-v geosig path, for any u,v € S1 and so that S1 is not a double signal set of G. It is easy to verify that S2 = S U {w1,w2, ...,wc-a+2} is a minimum double signal set of G and so dsn (G) = c.
Hence, sn(G) = g(G).
□
Vi V2
Va-2
Wc-a+2
Case 3. a < b = c. Let G be the graph in Fig. 6 got from the complete graph Kb-a+2 and the path P3 : x,y,z of order 3 by joining all the vertices of Kb-a+2 to x and y and adding a — 2 new pendant edges v1,v2, ...,va-2. By Theorem 2, Theorem 5 and Corollary 1, every geodetic set, every signal set and every double signal set of G contain the set S = {v1,v2,..., va-2} of all extreme vertices of G. Clearly, S is not a geodetic set of G. Also, for any u € V(G) — S, S U {u} is not a geodetic set or a signal set of G and so g(G) > a. Since S1 = S U {x, y} is a geodetic set of G, it follows that g(G) = a. It is clear that the vertices of Kb-a+2 do not lie on any signal path between vertices from S1, that S1 is not a signal set of G. Clearly, every signal set and every double signal set contains every vertices of Kb-a+2. Now it is easily to verify that S U V(Kb-a+2) is a minimum signal set and minimum double signal of G. Hence, sn (G) = dsn (G) = a — 2 + b — a + 2 = b.
X
Figure 6. Graph G.
Case 4. a <b < c. Let G be the graph in Fig. 7 obtained from the path P3 : x,y,z of order 3 by adding c new vertices u1,u2, ...,ua-2,w1,w2, ...,wb-a,v1 ,v2,...,vc-b+2 to P3 and joining each vertex wi(1 < i < b — a) to the vertices x,y and z ; joining each vertex vj(1 < j < c — b + 2) to the vertices x and z; joining each vertex uk(1 < k < a — 2) to the vertex y. By Theorems 2, Theorem 5 and Corollary 1, every geodetic set, every signal set and every double signal set of G contain the set S = {v1,v2, ...,va-2} of all extreme vertices of G. Clearly, S is not a geodetic set of G. Also, for any v V(G) — S, S U {v} is not a geodetic set of G and so g(G) > a. Since S1 = S U {x, z} is a geodetic set of G, it follows that g(G) = a. Since each vertex wj does not lie on any geosig of vertices of S1, that S1 is not a signal set of G. It is clear that every signal set of G contains {w1 ,w2, ...,wb-a}. Then S2 = S1 U {w1,w2, ...,wb-a} is a minimum signal set of G and so sn (G) = b. Now, each pair vj is either an initial vertex or terminal vertex of any signal path containing the vertices vj and wi. Hence v1,v2, ...,vc-b+2 are weak extreme vertices. It is easily verified that the set S3 = S U {w1 ,w2, ...,wb-a,v1 ,v2, ...,vc-b+2} is the unique minimum double signal set of G and so dsn (G) = c.
Theorem 15. For integers a,b and c with 3 < a < b < c, there exists a connected graph G with sn (G) = a, g(G) = b and dsn (G) = c.
Proof. This theorem is proved by considering three cases.
Case 1. a = b = c. Then for the complete graph Ka, g(G) = sn (G) = dsn (G) = a.
x
Case 2. a = b < c. The proof is similar to the proof of case 2 in Theorem 14.
Case 3. a < b < c. Let H be the graph obtained from the path P6 : u, v,w,x,y,z of order 6, b — a copies of path Pj : xj,yj,zj (1 < i < b — a) of order 3 and 2 copies of path Pj : xj, yj (1 < j < 2) of order 3 by joining each vertex xj(1 < i < b — a) to the vertex u of P6, each vertex Zj(1 < i < b — a) to the vertex v of P6, each vertex xj (1 < j < 2) to the vertex u of P6 and each vertex yj (1 < j < 2) to the vertex v of P6 and add an edge x1 y2. Let G be the graph in Fig. 8 obtained from H by adding the following new vertices to H.
(i) Add a — 1 new vertices u1, u2,..., ua-1 to H and join each uj(1 < i < a — 1) to w.
(ii) Add b — a — 1 new vertices v1, v2,..., vb-a-1 to H and join each vj(1 < i < b — a — 1) to both w and y.
(iii) Add c — b new vertices w1, w2,..., wc-b to H and join each Wj(1 < i < c — b) to both v and x.
Let S = {u1, u2,..., ua-1, z} be the set of extreme vertices of G. By Theorem 5, Theorem 2 and Corollary 1, every signal set, every geodetic set and every double signal set contains S. Clearly S itself is not a signal set of G and so sn (G) > a. It is clear that S1 = S U {u} is a signal set of G and hence sn (G) = a. Since the vertices Xj, yj, zj(1 < i < b — a) do not lie on any geodesic joining a pair of vertices from S1, S1 is not a geodetic set of G. Let S2 = S U {y1,y2, ...,yb-a}. It is easy to verify that S2 is a minimum geodetic set of G and so g(G) = a + b — a = b. Since the pair of vertices wj(1 < i < c — b),vj(1 < j < b — a — 1) do not lie on any signal path between a pair of vertices from S1, S1 is not a double signal set of G. Also, x is either an initial vertex or terminal vertex of any geosig containing the vertices x and v1 and so x is a weak extreme vertex. Hence w1, w2,..., wc-b, v1, v2,..., vb-a-1, x are weak extreme vertices.
Let S'= S1U{w1, w2,..., wc-b, v1, v2,..., vb-a-1 ,x}. It is easily verified that S' is the set of all weak extreme vertices of G. Since S' is a double signal set of G, by Theorem 6 it follows that dsn (G) = c. □
Theorem 16. For every pair a, b of integers with 4 < a < b and b = a + 1, there exists a connected graph G with dg(G) = a and dsn (G) = b.
Proof. For 4 < a = b, then the complete graph Ka has the desired properties. So, assume that 4 < a < b and b = a + 1. Let G be the graph in Fig. 9 formed from the path P4 : u, v, w, y of
order 4, by adding b new vertices u1,u2, ...,ua-3,v1 ,v2, ...,vb-a-1 ,x to P4 and joining each vertex ui(1 < i < a — 3) to v; and joining each vertex vj(1 < j < b — a — 1) to both v and y; and join the vertex x to u and w. Let S = {u1 ,u2, ...,ua-3} be the set of extreme vertices of G. By Theorem 3 and Corollary 1, every double geodetic and every double signal set contains S. Now it is clear that S1 = S U {u, x, y} is a minimum double geodetic set of G and so dg (G) = a. Since L[u,y] contains u,x,w,y, the pair of vertices x,vi(1 < i < b — a — 1) do not lie on any geosig of a pair of vertices from S. So that S1 is not a double signal set of G. It is easy to verify that
S2 = S U {v1,v2, ...,vb-a-1 ,w} be the unique minimum double signal set of G. Hence, dsn(G) = b.
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4. Closing open problems
We close with the following list of open problems that we have yet to settle. Problem 1. Determine the class of graphs G for which g(G) = sn(G). Problem 2. Determine the class of graphs G for which sn (G) = dsn (G).
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