Influence of edge subdivision on the convex domination number

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 53 (2012), Pages 19–30 Influence of edge subdivision on the convex domination number Magda Dettlaff Magdalena Lemańska Department of Technical Physics and Applied Mathematics Gdańsk University of Technology Narutowicza 11/12, 80–233 Gdańsk Poland [email protected] [email protected] Abstract We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G. 1 Introduction Let G = (V, E) be a connected undirected graph with |V | = n. The neighbourhood of a vertex v ∈ V in G is the set NG (v) of all vertices adjacent to v in G. For a set  X ⊆ V, the open neighbourhood NG (X) is defined to be v∈X NG (v) and the closed neighbourhood NG [X] = NG (X) ∪ X. The degree dG (v) = d(v) of a vertex v ∈ V is the number of edges incident to v; dG (v) = |NG (v)|. The minimum and maximum degrees among all vertices of G are denoted by δ(G) and ∆(G), respectively. A vertex u of degree d(u) = 1 we call an end-vertex. A support is a vertex adjacent to an end-vertex. A set of all end-vertices of a graph G we denote by Ω(G) and a set of all supports of G by S(G). If G is connected and δ(G) = ∆(G) = 2, then G is a cycle and the cycle on n vertices is denoted by Cn . The length of a shortest cycle in G is the girth of G and is denoted by g(G). A set D ⊆ V is a dominating set of G if NG [D] = V . The domination number of G, denoted γ(G), is the minimum cardinality of a dominating set in G. The distance dG (u, v) between two vertices u and v in a connected graph G is the length of the shortest (u−v) path in G. A (u−v) path of length dG (u, v) is called a (u−v)-geodesic. For unexplained terms and symbols see [6]. 20 MAGDA DETTLAFF AND MAGDALENA LEMAŃSKA A set X ⊆ V is convex in G if vertices from all (a − b)-geodesics belong to X for every two vertices a, b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number γcon (G) of a graph G is the minimum cardinality of a convex dominating set. The convex domination number was first introduced by Jerzy Topp, Gdańsk University of Technology (2002) and studied in [1], [7], [8]. The subdivision of some edge e = uv in a graph G yields a graph containing one new vertex w, and with an edge set replacing e by two new edges with endpoints uw and wv. Let us denote by Guv or Ge the graph obtained from a graph G by subdivision of an edge e = uv in G. The domination subdivision number sdγ (G) of a graph G is the minimum number of edges that must be subdivided, where each edge in G can be subdivided at most once, in order to increase the domination number. The domination subdivision number was defined in [10] and has been studied also in [3], [4]. The similar concept related to total domination was defined in [5]. Moreover, the weakly connected domination subdivision number was defined in [9] and studied in [2]. In this paper we study similar concepts related to convex domination. Since an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number (see Theorem 1), we do not define the subdivision number for convex domination, we just consider the influence of the edge subdivision on the convex domination number and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G. 2 Results We show that an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. Theorem 1 The difference between γcon (G) and γcon (Guv ) and between γcon (Guv ) and γcon (G) can be arbitrarily large. Proof. First we show that γcon (Guv ) − γcon (G) can be arbitrarily large. We show that for any positive integer k there exists a graph G such that γcon (Guv ) − γcon (G) = k. We shall construct a graph G in a following way: we begin with Kk−1 + K2 , where {u, v} are two vertices belonging to K2 . Then we add two vertices x, y and two edges xu, yv. It is easy to observe that {u, v} is a minimum convex dominating set of G and in such a graph G is γcon (G) = 2. If we subdivide an edge uv, then we obtain the graph Guv , where V (Guv )−Ω(Guv ) is a minimum convex dominating set of Guv . Thus γcon (Guv ) = k + 2 and γcon (Guv ) − γcon (G) = k. Figure 1 shows G and Guv for k = 4. Now we show that the difference γcon (G) − γcon (Guv ) can be arbitrarily large. We construct a graph G in a following way: we begin with a cycle C2k+2 with 2k + 2 vertices and label the vertices of the cycle consecutively v1 , . . . , vk , z, wk , wk−1 , . . . , w1 , u. 21 INFLUENCE OF EDGE SUBDIVISION Then we add the edges vi wi for i = 1, . . . , k and finally we add two vertices x, y and two edges xu, yz. Again it is easy to observe that in such a graph G, V (G) − Ω(G) is a minimum convex dominating set of G and γcon (G) = 2k + 2. The graph Guw1 we obtain by subdivision of an edge uw1 . In this graph, the minimum convex dominating set is the set of supports of Guw1 together with vertices v1 . . . , vk . Thus γcon (Guw1 ) = k + 2 and γcon (G) − γcon (Guw1 ) = k. Figure 2 shows the graphs G and Guw1 for k = 3. We have shown that in general the differences between γcon (G) and γcon (Guv ) and between γcon (Guv ) and γcon (G) can be arbitrarily large. But there are some classes of graphs where we can find some bounds. We begin with the following observation. Observation 1 If G is a unicyclic graph with the unique cycle C and if D is a minimum convex dominating set of G, then at most two vertices of C do not belong to D. Proof. Suppose to the contrary that three or more vertices of C do not belong to D; say c1 , . . . , ck do not belong to D, k ≥ 3. Since G is unicyclic and D is convex, c1 , . . . , ck are consecutive vertices. Since D is dominating, every ci has a neighbour in D. If two ci have the same neighbour in D, then C = C3 and k = 2, a contradiction. Thus every ci has a different neighbour in D. Let c′i ∈ NG (ci ) ∩ D, c′i+1 ∈ NG (ci+1 ) ∩ D, 1 ≤ i ≤ k − 1. Since D is convex, there is a (c′i − c′i+1 )-path in D, which produces a cycle in G. Thus c′i , c′i+1 ∈ C and every vertex of C except ci , ci+1 belongs to D. Theorem 2 If G is a unicyclic graph with the only cycle C, then γcon (G) ≤ γcon (Guv ) ≤ γcon (G) + 3. tu ✚ ✡ ✡ ❏ ✚✚ ❏ ✡✡ ✚ ✚ t ❏ ❩ ✡❏ ❩ ✡ ❏ ✡ ❩❩ ❏❏t t ✡ ❩ t ❏ v x t t y u t t ✚ ❏ ✡ ✡ ❏ ✚✚ ❏ ✡ ✚ t t✚ ❏✡ ❩ ✡❏ ❩ ✡ ❏ ✡ ❩❩ ❏❏t ✡ t ❩ v Figure 1: Graphs G and Guv for k = 4. x t t y 22 MAGDA DETTLAFF AND MAGDALENA LEMAŃSKA tx tx u ✟t❍ ❍❍ w1 ❍t
❆
❆
❆t
t ❆
w2 v2❆❆
❆
❆ t t
v3❍❍❍ ✟ w3 ❍t✟✟✟ ✟ v1 ✟ t ✟ z ty u ✟t❍ tw ❍ ❍ w1 ❍t
❆
❆
❆t t
❆
w2 v2❆❆
❆
❆ t t
v3❍❍❍ ✟ w3 t✟✟ ❍✟ v1 t✟✟✟ z ty Figure 2: Graph G and Guw1 for k = 3. Proof. First we show that γcon (Guv ) ≤ γcon (G) + 3. Let D be a minimum convex dominating set of G and let uv be the subdivided edge. We consider three cases. Case 1. Let u ∈ D and v ∈ D. If uv is not a cyclic edge, then D ∪ {w} is a convex dominating set of Guv and γcon (Guv ) ≤ |D| + 1 = γcon (G) + 1 ≤ γcon (G) + 3. Now let uv ∈ E(C). From Observation 1, at most two vertices of C do not belong to D. If every vertex of C belongs to D, then D ∪ {w} is a convex dominating set of Guv and we are done. Now let one vertex, say z, of C belong to V − D. Since D is convex, C = C3 . Then D ∪ {w, z} is a convex dominating set of Guv and again we obtain the required inequality. If two vertices, let us say, z1 , z2 of a cycle C belong to V − D, then, since D is convex, C = C4 or C = C5 . If C = C4 , then D ∪ {w} is a convex dominating set of Guv ; if C = C5 , then D ∪ {w, z1 , z2 } is a convex dominating set of Guv and we are done. Case 2. Now let |{u, v} ∩ D| = 1; without loss of generality let u ∈ D, v ∈ V − D. If e = uv does not belong to C, then similarly to the previous case, D ∪ {w} is a convex dominating set of Guv and we obtain the desired inequality. If e = uv belongs to the cycle C and C = C3 , then by Observation 1, two vertices or one vertex of the cycle are outside D. If exactly one vertex v of the cycle C3 is outside D, then D is a convex dominating set of Guv and γcon (Guv ) ≤ |D| = γcon (G) ≤ γcon (G) + 3. If two vertices of C3 do not belong to D, then D ∪ {w} is a convex dominating set of Guv and we are done. Now let e = uv belong to the cycle C = C4 . From Observation 1, at most two vertices of the cycle are outside D. Since D is convex, no vertices or else exactly two vertices of C are in V − D, and then D ∪ {w} is a convex dominating set of Guv and we obtain the required inequality. If e = uv belongs to the cycle C5 , then, since D is convex, no vertices from C5 or exactly two vertices of C5 belong to V − D. If no vertex of C5 belongs to V − D, then D ∪ {w} is a convex dominating set of Guv and we obtain the required inequality. If exactly two vertices of C5 belong to V − D, let us say v, x ∈ (V − D) ∩ C5 , then D ∪ {w, v, x} is a convex dominating set of Guv and γcon (Guv ) ≤ γcon (G) + 3. INFLUENCE OF EDGE SUBDIVISION 23 If e = uv belongs to the cycle Cp , p ≥ 6, then, since D is convex, every vertex of Cp belongs to D, which gives a contradiction with the fact that |{u, v} ∩ D| = 1. Case 3. Now let u ∈ V − D and v ∈ V − D. Then, since u and v are dominated and since D is convex, both vertices u, v belong to C. Since D is a convex set and u, v ∈ / D, we have C = Cp , 3 ≤ p ≤ 5. If p = 3 or p = 4, then D ∪ {u} is a convex dominating set of Guv and we are done. If p = 5, then D ∪ {u, v, w} is a convex dominating set of Guv and finally we obtain the required inequality. Now we show that γcon (G) ≤ γcon (Guv ). Let D0 be a minimum convex dominating set of Guv . Again we consider three cases. Case 1. Let u ∈ D0 , v ∈ D0 . Then D0 − {w} is a convex dominating set of G and thus γcon (G) ≤ γcon (Guv ) − 1, which gives γcon (G) ≤ γcon (G) + 1 ≤ γcon (Guv ). Case 2. Now let u ∈ D0 , v ∈ / D0 . If w ∈ D0 , then D0 − {w} is a convex dominating set of G and we are done. If w ∈ / D0 , then D0 is a convex dominating set of G and γcon (G) ≤ γcon (Guv ). / D0 , then u, v, w must have neighbours in D0 , which Case 3. If u ∈ / D0 and v ∈ produces more than one cycle, a contradiction. Corollary 3 Let G be a unicyclic graph with the only cycle C and let D be a minimum convex dominating set of G. • If e is not a cyclic edge, then γcon (Ge ) = γcon (G) + 1. • If V (C) ⊆ D, then for any edge e ∈ E(G) we obtain γcon (Ge ) = γcon (G) + 1. • If there is v ∈ V (C) such that v ∈ D, then C = C3 or C = C4 or C = C5 : – If C = C3 and |V (C) ∩ D| = 1, then for any edge e ∈ E(G) we obtain γcon (Ge ) = γcon (G) + 1. If |V (C) ∩ D| = 2, let us say x, y, z ∈ V (C) and x, y ∈ D, then γcon (Ge ) = γcon (G) + 1 for e = xy and γcon (Ge ) = γcon (G) for e = xz and e = yz. – If C = C4 , then |V (C) ∩ D| = 2 and for any edge e ∈ E(G) we obtain γcon (Ge ) = γcon (G) + 1. – If C = C5 , then |V (C) ∩ D| = 3 and for any edge e ∈ V (C) we obtain γcon (Ge ) = γcon (G) + 3. Now we investigate graphs G for which the convex domination number increases by exactly one after subdividing an edge. Proposition 4 If T is a tree of order at least three, then for any edge uv of T is γcon (Tuv ) = γcon (T ) + 1. 24 MAGDA DETTLAFF AND MAGDALENA LEMAŃSKA Proof. The only minimum convex dominating set of a tree T is D = V (T ) − Ω(T ). After subdividing any edge we obtain the tree T ′ such that |V (T ′ )| = |V (T )| + 1 and |Ω(T ′ )| = |Ω(T )|. Hence we have γcon (Tuv ) = γcon (T ) + 1 for any edge uv of T . Observation 2 Let G be a connected graph with δ(G) = 1 and let uv be an end-edge of G. Then γcon (Guv ) = γcon (G) + 1. Observation 3 Let G be a connected graph with ∆(G) = n(G) − 1. Then γcon (Guv ) = γcon (G) + 1 for any edge uv ∈ E(G). Theorem 5 [1] If G is a connected graph with δ(G) ≥ 2 and g(G) ≥ 6, then γcon = n(G). Since subdividing an edge does not decrease the girth of the graph, we have the following corollary. Corollary 6 If G is a connected graph with δ(G) ≥ 2 and g(G) ≥ 6, then γcon (Guv ) = γcon (G) + 1 for any edge uv ∈ E(G). Theorem 7 If G is a connected graph with g(G) ≥ 6, then γcon (G) = n − |Ω(G)|. Proof. If δ(G) ≥ 2, then the result holds by Theorem 5. Let δ(G) = 1. Let D be a minimum convex dominating set of G. Of course, no end-vertex belongs to D. Let v be any non-end-vertex of G. If v does not belong to a cycle, then v belongs to some (a − b)-geodesic, where we have the following possibilities for vertices a, b: • a ∈ V (C1 ) and b ∈ V (C2 ), where C1 , C2 are cycles in G; • a, b ∈ S(G); • a ∈ V (C) and b ∈ S(G), where C is a cycle in G. Thus v ∈ D. Now let v belong to a non-induced cycle C with p vertices; of course p ≥ 6. Suppose v ∈ / D. The vertex v has k neighbours v1 , . . . , vk , k ≥ 2. Suppose more than one of vi , 1 ≤ i ≤ p belongs to D; without loss of generality let v1 , v1 ∈ D. Since g(G) ≥ 6, we have v1 v2 ∈ / E(G), |NG (v1 ) ∩ NG (v2 )| = 1 and v is the only vertex belonging to NG (v1 ) ∩ NG (v2 ). But then v belongs to a (v1 − v2 )-geodesic and v ∈ D, a contradiction. Thus v has exactly one neighbour in D; let v1 ∈ NG (v) ∩ D. Then v2 ∈ V − D. Since v2 is dominated, there exists y ∈ D ∩ NG (v2 ), y = v1 , since g(G) ≥ 6. Since D is convex, vertices from every (v1 − y)-geodesic belong to D. Since v1 v ∈ E(G), vv2 ∈ E(G) and v2 y ∈ E(G), a (v1 −y)-geodesic has length at most 3. If it has length INFLUENCE OF EDGE SUBDIVISION 25 one or two, we obtain a cycle of length less than 6, a contradiction. If it has length 3, then v, v2 belong to a (v1 − y)-geodesic and v, v2 ∈ D, a contradiction. Thus v ∈ D. Then |D| ≥ n − |Ω(G)|. On the other hand, V (G) − Ω(G) is a convex dominating set of G and thus γcon (G) ≤ |V (G) − Ω(G)| = n − |Ω(G)|. Corollary 8 If g(G) ≥ 6, then γcon (Guv ) = γcon (G) + 1. Proof. Let e be a subdivided edge and let D′ be a minimum convex dominating set of Guv . If e belongs to a cycle C, then g(Guv ) ≥ 6 and |D′ | = n(Guv ) − |Ω(Guv )| = n(G) + 1 − |Ω(G)| = γcon (G) + 1. If e does not belong to C, then the new vertex w of Guv lies on an (a − b)geodesic, where a, b belong to a cycle or to S(G). So x ∈ D′ and |D′ | = |D| + 1, where |D| = |V (G) − Ω(G)| and γcon (Guv ) = γcon (G) + 1. For an edge e = uv ∈ E(G), let us define diff(e) = γcon (Guv ) − γcon (G) and for a graph G we consider S ′ (G) = e∈E(G) diff(e). Now we show that for every integer k there exists a graph G such that S ′ (G) = k. We begin with the definition of the family of graphs G. Let G be the family of graphs G that can be obtained from a sequence G1 , . . . , Gj (j ≥ 1) of graphs such that G1 is a graph shown in Figure 3 and G = Gj , and, if j ≥ 2, then Gi+1 can be obtained from Gi by operation Y listed below. We define the status of a vertex v denoted sta(v) to be A or B, where initially for G1 we put sta(v) = A if v is an end-vertex or a support of G1 and sta(v) = B if v is a vertex of degree two in G1 . Once a vertex is assigned a status, this status remains unchanged as the graph is recursively constructed. • Operation Y. The graph Gi+1 is obtained from Gi by adding the graph H shown in Figure 3 and identifying an end-vertex of Gi with a vertex u of H. Then we let sta(u) = A, sta(x) = A if x ∈ V (H) is a support vertex or an end-vertex and sta(y) = B if y is a vertex of degree two in H and y = u. If Gk ∈ G, k ≥ 1, is a graph obtained by using k − 1 times of operation Y, then by Gtk , t ≥ 0, we denote a graph obtained from Gk by adding t end-vertices u1 , . . . , ut and t pendant edges wk u1 , . . . , wk ut , where wk is a support vertex of Gk . In particular G0k = Gk . In Figure 4 we have Gtk for t = 2 and k = 2. Every vertex ui is assigned a status A. The edge e = uv we call an (A − A)-edge if sta(u) = A and sta(v) = A. In the other case we denote it (A − B)-edge. Observation 4 For a graph Gk ∈ G we have: • diff(e) = 1 if e is (A − A)-edge; 26 MAGDA DETTLAFF AND MAGDALENA LEMAŃSKA t ❅ ❅ t t ❅ ❅ ❅t ❅t t t w1 t ❅ ❅ ❅t u❅ v1 t ❅ ❅t H G1 Figure 3: Graphs G1 and H t ❅ ❅ t t ❅ ❅ ❅t t ❅ ❅ t ❅ ❅t ❅ ❅t t u1 ❅t w❅ 2 ❅ t v2 ❅t u2 Figure 4: Graph G22 • diff(e) = −1 if e is (A − B)-edge. Lemma 9 For every integer k there exists a graph G such that  S ′ (G) = diff(e) = k. e∈E(G) Proof. By Observation 4 we have that S ′ (G1 ) = −2 and S ′ (Gt1 ) = −2 + t, where t ≥ 0. Hence for k ≥ −2 there exists a graph G = Gk+2 such that S ′ (G) = k. 1 For k ≤ −3 we consider three cases: Case 1. Let k ≡ 1 (mod 3). This gives k = 3p + 1 for an integer p < −1. Thus for G = G−p we have S ′ (G) = 3p + 1 = k. Case 2. Now let k ≡ 2 (mod 3). This gives k = 3p + 2 for an integer p < −1. Thus for G = G1−p we have S ′ (G) = 3p + 2 = k. Case 3. If k ≡ 0 (mod 3). This gives k = 3p for an integer p < 0. Thus for G = G2−p+1 we have S ′ (G) = 3p = k. Lemma 10 For any integer k ≥ 3 there exists a graph G such that for any edge e ∈ E(G) we have diff(e) = k. INFLUENCE OF EDGE SUBDIVISION ut1 ut2 ut3 ut4 t t t t v1 v2 v3 v4 27 utk−3 utk−2 utk−1 ❅ t t t ❅ ❅t vk−3 vk−2 vk−1 vk Figure 5: Graph H k w1✟✟ t❆ ✏ t❣ ✏ t✏
❆ ❆t t❣

❆❣ t✟✟ ❆ t

❆ ❆t
❍❍
t❣ t✟✟ ❆❣
t t
❍❍
t❣ t✟✟ ❆ ❣ ❆ ❆t t ❆❣ t✟✟ ❆
❍❍t❣ ❆ ❆t

t ❆❣ t ✂

❍❍t❣ ❆ ✂

t ❍❍
t❣ t❣ t❣ t❣ ❣ t✂ w2 t ❣ t ❆❣ w3❅ ❅t t t t t t Figure 6: Graph Gk Proof. We show the example of a graph such that subdivision of any edge of this graph increases the convex domination number by k ≥ 3. First, let us consider a graph H k , k ≥ 3, constructed in the following way. We begin with a path Pk = (v1 , v2 , . . . , vk ) and a path Pk−1 = (u1 , u2 , . . . , uk−1 ) and then we add edges u1 v1 , u2 v2 , . . . , uk−2 vk−2 , uk−1 vk (see Fig. 5). Next, we take three copies of H k : H k1 , H k2 and H k3 . For j = 1, 2, 3 we denote vertices of H kj by v1j , v2j , . . . , vkj , uj1 , uj2 , . . . , ujk−1 . Afterwards, we take the union of graphs H k1 , H k2 and H k3 and identify vertices v11 and vk3 (which gives a vertex w1 ), vertices vk1 and v12 (which gives a vertex w2 ) and vertices vk2 and v13 (what gives a vertex w3 ). In this way we obtain a graph Gk (see Fig. 6). Note that the graph Gk can be obtained also from a cycle C3 with vertices w1 , w2 and w3 by an adequate replacement the edges of C3 with the copy of H k . Instead of C3 we can also consider cycle Cp with vertices w1 , . . . , wp for p ≥ 4 to obtain more general example of G. 1 In Gk , vertices w1 , w2 and w3 and vertices belonging to (v21 − vk−1 ), (v22 − 2 3 vk−1 ), (v23 − vk−1 )-geodesics create the unique minimum convex dominating set (see Figure 6). Hence, γcon (Gk ) = 3k − 3. Because of the symmetry of the graph Gk , it suffices to consider subdivision of the edge from one copy of H k , let us say H 1 . We denote the minimum convex dominating set of Gk with subdivided edge e, Gke , by D′ ′ 2 3 and, because of the convexity of D , vertices w1 , w2 , w3 and v22 , . . . , vk−1 , v23 , . . . , vk−1 ′ belong to D . Let e be the subdivided edge. We consider four cases: 28 MAGDA DETTLAFF AND MAGDALENA LEMAŃSKA Case 1. First let e = vi1 vj1 , 1 ≤ i, j ≤ k. Then dGe (v11 , vk1 ) = k + 1 and ver1 tices w, v21 , v31 , . . . , vk−1 , u11 , u12 , . . . u1k−1 belong to (v11 − vk1 )-geodesic, so they are in 1 D′ . Hence, |D′ | ≥ 4k − 3. On the other hand, {w1 , w2 , w3 , w, v21 , v31 , . . . , vk−1 , 1 1 1 2 2 3 3 k u1 , u2 , . . . uk−1 , v2 , . . . , vk−1 , v2 , . . . , vk−1 } is a convex dominating set of Ge . Hence, |D′ | ≤ 4k − 3. Finally, diff(vi1 vj1 ) = |D′ | − |D| = 4k − 3 − (3k − 3) = k. 1 Case 2. Now let e = v11 u11 . Then dGe (v11 , vk1 ) = k and v11 , v21 , v31 , . . . , vk−1 , vk1 ∈ 1 ′ ′ 1 D . Because k ≥ 3, w ∈ D in order to dominate u1 and dGe (w, vk ) = k + 1. Hence, u11 , u12 , . . . , u1k−1 belong to a (w − vk1 )-geodesic, so they are in D′ . This gives 1 |D′ | ≥ 4k − 3. On the other hand, {w1 , w2 , w3 , w, v21 , v31 , . . . , vk−1 , u11 , u12 , . . . u1k−1 , 2 3 v22 , . . . , vk−1 , v23 , . . . , vk−1 } is a convex dominating set of Gke . Hence, |D′ | ≤ 4k − 3. Finally, diff(v11 u11 ) = |D′ | − |D| = 4k − 3 − (3k − 3) = k. Similarly we can show that diff(u1k−1 vk1 ) = k. 1 Case 3. Let e = u1i u1j , 1 ≤ i, j ≤ k − 1. Vertices v11 , v21 , v31 , . . . vk−1 , vk1 ∈ D′ and w is not dominated. Case 3.1. Let u1i ∈ D′ . Then dGe (v11 , u1i ) = i and vertices u11 , u12 , . . . , u1i−1 belong to a (u1i − v11 )-geodesic, so they are in D′ . Moreover, dGe (u1i , vk1 ) = k − i + 1 and w, u1j , u1j+1 , . . . , u1k−1 belong to a (u1i −vk1 )-geodesic, which implies they are in D′ . Then 1 |D′ | ≥ 4k − 3. On the other hand, {w1 , w2 , w3 , w, v21 , v31 , . . . , vk−1 , u11 , u12 , . . . u1k−1 , 2 3 v22 , . . . , vk−1 , v23 , . . . , vk−1 } is a convex dominating set of Gke . Hence, |D′ | ≤ 4k − 3. Finally, diff(u1i u1j ) = |D′ | − |D| = 4k − 3 − (3k − 3) = k. Case 3.2. Let u1j ∈ D′ . Then dGe (v11 , u1j ) = j and vertices u11 , u12 , . . . , u1i , w belong to a (u1j − v11 )-geodesic, so they are in D′ . Moreover, dGe (u1j , vk1 ) = k − j + 1 and u1j+1 , u1j+2 , . . . , u1k−1 belong to a (u1j −vk1 )-geodesic, which implies they are in D′ . Then 1 |D′ | ≥ 4k − 3. On the other hand, {w1 , w2 , w3 , w, v21 , v31 , . . . , vk−1 , u11 , u12 , . . . u1k−1 , 2 2 3 3 k v2 , . . . , vk−1 , v2 , . . . , vk−1 } is a convex dominating set of Ge . Hence, |D′ | ≤ 4k − 3. Finally, diff(ui uj ) = |D′ | − |D| = 4k − 3 − (3k − 3) = k. 1 Case 4. If k ≥ 4, we subdivide an edge u1i vi1 , 2 ≤ i ≤ k−2. Vertices v11 , v21 , v31 , . . . vk−1 , 1 ′ 1 vk ∈ D and ui is not dominated. Case 4.1. Let w ∈ D′ . Then dGe (w, vk1 ) = k − i + 1 and vertices u1i , u1i+1 , . . . , u1k−1 belong to a (w − vk1 )-geodesic, so they are in D′ . Moreover, dGe (u1i , v11 ) = i + 1 and u11 , u12 , . . . , u1i−1 belong to (u1i − v11 )-geodesic, which implies they are in D′ . This gives 1 |D′ | ≥ 4k − 3. On the other hand, {w1 , w2 , w3 , w, v21 , v31 , . . . , vk−1 , u11 , u12 , . . . u1k−1 , 2 3 v22 , . . . , vk−1 , v23 , . . . , vk−1 } is a convex dominating set of Gke . Hence, |D′ | ≤ 4k − 3. Finally, diff(ui vi ) = |D′ | − |D| = 4k − 3 − (3k3 ) = k. Case 4.2. Let u1i+1 ∈ D′ . Then dGe (v11 , u1i+1 ) = i + 1 and vertices u11 , u12 , . . . , u1i belong to a (u1i+1 − v11 )-geodesic, so they are in D′ . Moreover, dGe (u1i+1 , vk1 ) = k − i − 1 and u1i+2 , u1i+3 , . . . , u1k−1 belong to a (u1i+1 −vk1)-geodesic, which implies they are in D′ . Also w ∈ D′ , because it belongs to a (u1i − vi1 )-geodesic. This gives |D′ | ≥ 4k − 3. On the 29 INFLUENCE OF EDGE SUBDIVISION 1 2 3 other hand, {w1 , w2 , w3 , w, v21 , v31 , . . . , vk−1 , u11 , u12 , . . . u1k−1 , v22 , . . . , vk−1 , v23 , . . . , vk−1 } is a convex dominating set of Gke . Hence, |D′ | ≤ 4k − 3. Finally diff(ui vi ) = |D′ | − |D| = 4k − 3 − (3k − 3) = k. Case 4.3. Let u1i−1 ∈ D′ . Similarly as in Case 4.2 we can show that diff(u1i vi1 ) = k. In Figure 7 we have an example of a graph G such that any edge e ∈ E(G) satisfies diff(e) = −1. By the symmetry of G it suffices to subdivide only one edge. Minimum convex dominating sets of G and Ge are indicated in Figure 7. t❍ ✟❣ ❍❍ ✟ ✟ ❍❍ ✟✟ ❍❣ ❣ ❣ t✟ t t ❅ ❅ ❣ t t ❅❣ ❅ ❅ ❣ ❣ t t ❅t❣ t❍ ✟❣ ❍❍ ✟ ✟ ❍❍ ✟✟ ❍❣ ❣ t✟ t t t ❅ ❅ ❣ t t ❅❣ ❅ ❅ ❣ ❣ t t t ❅❣ G Ge Figure 7: Graphs G and Ge Proposition 4 gives an example of graphs for which for any edge e satisfies diff(e) = 1. The existence of graphs G for which for any k < −1 and k = 0, 2 satisfies diff(e) = k remains an open problem. References [1] J. Cyman, M. Lemańska and J. Raczek, Graphs with convex domination number close to their order, Discuss. Math. Graph Theory 26 (2006), 307–316. [2] M. Dettlaff and M. Lemańska, Strong weakly connected domination subdivisible graphs, Australas. J. Combin. 47 (2010), 269–277. [3] O. Favaron, T. W. Haynes and S. T. Hedetniemi, Domination subdivision numbers in graphs, Util. Math. 66 (2004), 195–209. [4] T. W. Haynes, S. M. Hedetniemi, S. T. Hedetniemi, J. Knisely and L. C. van der Merve, Domination subdivision numbers, Discuss. Math. Graph Theory 21 (2001), 239–253. [5] T. W. Haynes, S. T. Hedetniemi and L. C. van der Merve, Total domination subdivision numbers, J. Combin. Math. Combin. Comput. 44 (2003), 115–128. 30 MAGDA DETTLAFF AND MAGDALENA LEMAŃSKA [6] T. Haynes, S. Hedetniemi and P. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc. (1998). [7] M. Lemańska, Weakly convex and convex domination numbers, Opuscula Mathematica 24/2 (2004), 181–188. [8] J. Raczek, NP-completeness of weakly convex and convex dominating set decision problems, Opuscula Mathematica 24 (2004), 189–196. [9] J. Raczek, Weakly connected domination subdivision numbers, Discuss. Math. Graph Theory 28 (2008), 109–119. [10] S. Velammal, Studies in Graph Theory: Covering, Independence, Domination and Related Topics, Ph.D. Thesis, Manonmaniam Sundaranar University, Tirunelveli, 1997. (Received 29 Dec 2010; revised 11 Jan 2012, 21 Mar 2012)