Coronas and Domination Subdivision Number of a Graph

Bulletin of the Malaysian Mathematical Sciences Society, Sep 2016

In this paper, for a graph G and a family of partitions \(\mathcal {P}\) of vertex neighborhoods of G, we define the general corona \(G \circ \mathcal {P}\) of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.

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Coronas and Domination Subdivision Number of a Graph

Coronas and Domination Subdivision Number of a Graph M. Dettlaff 0 1 M. Lema n´ska 0 1 J. Topp 0 1 P. Z˙ yli n´ski 0 1 0 Faculty of Mathematics , Physics and Informatics , University of Gdan ́sk , 80-952 Gdan ́sk , Poland 1 Faculty of Applied Physics and Mathematics, Gdan ́sk University of Technology , 80-233 Gdan ́sk , Poland In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G ◦ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3. Mathematics Subject Classification 05C69 · 05C05 · 05C99 Domination; Domination subdivision number; Tree; Corona - 1 Introduction In this paper, we follow the notation and terminology of [ 7 ]. Let G = (V (G), E (G)) be a (finite, simple, undirected) graph of order n = |V (G)|. For a vertex v of G, its Communicated by Xueliang Li. neighborhood, denoted by NG (v), is the set of all vertices adjacent to v. The cardinality of NG (v), denoted by dG (v), is called the degree of v. A vertex v is a leaf of G if dG (v) = 1. Every neighbor of a leaf is called a support vertex. A strong support vertex is a vertex adjacent to at least two leaves. A subset D of V (G) is said to be dominating in G if every vertex belonging to V (G) − D has at least one neighbor in D. The cardinality of the smallest dominating set in G, denoted by γ (G), is called the domination number of G. A subset S of vertices in G is called a 2-packing if every two distinct vertices belonging to S are at distance greater than 2. The corona of graphs G1 and G2 is a graph G = G1 ◦ G2 resulting from the disjoint union of G1 and |V (G1)| copies of G2 in which each vertex v of G1 is adjacent to all vertices of the copy of G2 corresponding to v. For a graph G, the subdivision of an edge e = uv with a new vertex x is an operation which leads to a graph G with V (G ) = V (G) ∪ {x } and E (G ) = (E (G) − {uv}) ∪ {ux , x v}. The graph obtained from G by the replacing every edge e = uv with a path (u, x1, x2, v), where x1 and x2 are new vertices, is called the 2-subdivision of G and is denoted by S2(G). For a graph G and a family P = {P(v): v ∈ V (G)}, where P(v) is a partition of the set NG (v), by G ◦ P, we denote the graph in which V (G ◦ P) = {(v, 1): v ∈ V (G)} ∪ {(v, A): A ∈ P(v)} v∈V (G) and E (G ◦ P) = {(v, 1)(v, A): A ∈ P(v)} v∈V (G) ∪ {(v, A)(u, B): (u ∈ A) ∧ (v ∈ B)}. uv∈E(G) The family P is called a vertex neighborhood partition of G and the graph G ◦ P is called a P-corona (or shortly general corona) of G. The set {(v, 1): v ∈ V (G)} of vertices of G ◦ P is denoted by E x t (G ◦ P) and its elements are called the external vertices. Those vertices of G ◦ P which are not external, are said to be internal. Example 1 Let G be the graph shown in Fig. 1a and let P = {P(a), P(b), P(c), P(d), P(e)}, where P(a) = {NG (a)} = {{b, c}}, P(b) = {NG (b)} = {{a, c, d, e}}, P(c) = {NG (c)} = {{a, b}}, P(d) = {NG (d)} = {{b}}, P(e) = {NG (e)} = {{b}}. Then the P-corona G ◦ P is the graph G1 given in Fig. 1b and in fact it is the corona G ◦ K1. Now if P = {P(v): v ∈ V (G)} and P(v) is the family of all 1-element subsets of NG (v), that is P(a) = {{b}, {c}}, P(b) = {{a}, {c}, {d}, {e}}, P(c) = {{a}, {b}}, P(d) = {{b}}, P(e) = {{b}}, then G ◦ P is the graph G2 shown in Fig. 1c and in this case it is the 2-subdivision S2(G) of G. Finally, let us consider—for an example—the case where P = {P(v): v ∈ V (G)} and P(a) = {{b, c}}, P(b) = {{a}, {c, e}, {d}}, P(c) = {{a}, {b}}, P(d) = {{b}}, (a) c e a (b) (c, 1) (c, {a, b}) (a, 1) (b, 1) (e, 1) (e, {b}) (d, 1) G1 = G ◦ K1 (a, {b, c}) (b, {a, c, d, e}) (d, {b}) b (c) G d (c, 1) (c, {a}) (a, {c}) (b, {c}) (d) (c, 1) (c, {a}) (a, {b, c}) (a, 1) (c, {b}) (c, {b}) (e, 1) (e, {b}) (b, {e}) (e, 1) (e, {b}) (b, {c, e}) (b, {a}) (b, 1) (b, {d}) (d, {b}) (d, 1) G2 = Sa(G) G3 (a, 1) (a, {b}) (b, {a}) (b, 1) (b, {d}) (d, {b}) (d, 1) P(e) = {{b}}. In this case, G ◦ P is the graph G3 shown in Fig. 1d. This graph is an example of possible general coronas of G which are “between” the corona G ◦ K1 and the 2-subdivision S2(G). From the definition of general corona, it obviously follows (as we have seen in the above example) that (a) if P(v) = {N (v)} for every v ∈ V (G), then G ◦ P is the corona G ◦ K1 (and the vertices of G are internal vertices in G ◦ K1); (b) if P(v) = {{u}: u ∈ NG (v)} for every v ∈ V (G), then G ◦ P is the 2-subdivision S2(G) (and the vertices of G are external vertices of S2(G)). Let H be a subgraph of a graph G. The contraction of H to a vertex is the replacement of H by a single vertex k. Each edge that joined a vertex v ∈ V (G) − V (H ) to a vertex in H is replaced by an edge with endpoints v and k. Let P = {P(v): v ∈ V (G)} and P = {P (v): v ∈ V (G)} be two vertex neighborhood partitions of G. We say that P is a refinement of P and write P ≺ P if for every v ∈ V (G) and every A ∈ P (v) there exists B ∈ P(v) such that A ⊆ B. If P ≺ P, then the general corona G ◦ P is said to be refinement of G ◦ P. In this case, we write G ◦ P ≺ G ◦ P and say that G ◦ P has been obtained from G ◦ P by splitting some of its internal vertices. On the other hand, G ◦ P can be obtained from G ◦ P contracting some of its internal vertices. For example, G2 from Fig. 1 is refinement of G3 and G3 is refinement of G1, so G2 ≺ G3 ≺ G1. Notice that in general, a graph G ◦ P can be treated as a graph obtained from corona G ◦ K1, where we split every support vertex v according to the partition P(v) of NG (v). Let us again consider the graphs G, G1, G2 and G3 from Fig. 1. The graph G2 = S2(G) can be obtained from G ◦ K1 by splitting support vertex into maximum possible number of vertices. Moreover, if in G ◦ K1 we split the vertex (c, {a, b}) into two vertices: (c, {a}) and (c, {b}), the vertex (b, {a, c, d, e}) into three vertices: (b, {a}), (b, {c, e}), (b, {d}), and we leave other support vertices unchanged, then we obtain G3. On the other hand, G3 can be obtained from G2 = S2(G) contracting (a, {c}) and (a, {b}), and also (b, {c}) and (b, {e}) The contraction (splitting) of internal vertices is called an internal contraction (splitting). We have the following observations: Observation 2 Let T be a tree with at least three vertices. Then, the following properties are equivalent: 1. T is a general corona of a tree. 2. There exists a tree T such that T is obtained from the 2-subdivision S2(T ) by a sequence of internal contractions. 3. There exists a tree T such that T is obtained from the corona T ◦ K1 by a sequence of internal splittings. Observation 3 If G is a general corona of a tree, then E x t (G) is a dominating 2packing of G containing all leaves of G. Proof It follows from the following three facts: The distance between any two external vertices of G is at least three. Next, every internal vertex of G is adjacent to an external vertex. Finally, every leaf of G belongs to E x t (G). Observation 4 Let G and H be general coronas of some trees. If they share only one vertex which is an external vertex in each of them, then G ∪ H is a general corona. Proof Assume that G and H are general coronas of some trees T1 and T2, say G = T1 ◦ P1 and H = T2 ◦ P2 for some neighborhood partitions P1 and P2 of T1 and T2, respectively. Let (v, 1) be the only common external vertex of G and H . Then the trees T1 and T2 share only v and the union T = T1 ∪ T2 is a tree. Now, let P be the family {P(x ): x ∈ V (T )}, where P(v) = P1(v) ∪ P2(v), P(x ) = P1(x ) for x ∈ V (T1) − {v}, and P(x ) = P2(x ) for x ∈ V (T2) − {v}. Then G ∪ H is a P-corona of T , that is, G ∪ H = T ◦ P, see Fig. 2. Observation 5 Let G be a general corona of a tree and let (v, 1) be an external vertex of G. If we contract two distinct neighbors of (v, 1), then the resulting graph is also a general corona of a tree. Proof Assume that G = T ◦ P for some tree T and its neighborhood partition P. Let (v, A) and (v, B) be distinct neighbors of (v, 1). Then the graph G , obtained from G by the contraction of (v, A) and (v, B), is a P -corona of T , where P (v) = (P(v) − { A, B}) ∪ { A ∪ B}, and P (x ) = P(x ) if x ∈ V (T ) − {v}, see Fig. 3. From Observations 4 and 5, we immediately have the next observation (see Fig. 4 for an illustration). Observation 6 Let G and H be general coronas of some trees. If they share only one edge such that exactly one of its end vertices is an external vertex in each of G and H , then the union G ∪ H is a general corona. 2 Trees with Domination Subdivision Number 3 The domination subdivision number of a graph G, denoted by sd(G), is the minimum number of edges which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Since the domination number of the graph K2 does not increase when its edge is subdivided, we consider the subdivision numbers for connected graphs of order at least 3. The domination subdivision number was defined by Velammal [ 8 ] and since then it has been widely studied, see [ 2–6 ] to mention just a few. It was shown in [ 8 ] that the domination subdivision number of a tree is either 1, 2, or 3. Let Si be the family of trees with domination subdivision number equal to i for i ∈ {1, 2, 3}. Some characterizations of the classes S1 and S3 were given in [ 2 ] and [ 1 ], respectively. In particular, the following constructive characterization of S3 was given in [ 1 ]. Let the label of a vertex v be denoted by l(v) and l(v) ∈ { A, B}. Now, let F be the family of labeled trees that (i) contains P4, where leaves have label A and support vertices have label B, and (ii) is closed under the following two operations, which extend a labeled tree T ∈ F by attaching a labeled path to a vertex v ∈ V (T ) in such a way that: • If l(v) = A, then we add a path (x , y, z) (with labels l(x ) = l(y) = B and l(z) = A) and an edge vx . • If l(v) = B, then we add a path (x , y) (with labels l(x ) = B and l(y) = A) and an edge vx . The following characterization of trees belonging to the class S3 was given in [ 1 ]. Theorem 7 The next three statements are equivalent for a tree T with at least three vertices: 1. T belongs to the class S3. 2. T has a unique dominating 2-packing containing all leaves of T . 3. T belongs to the family F . Now we are in position to give a new characterization of trees belonging to the class S3. Namely, we shall show that all these graphs precisely are general coronas of trees. Lemma 8 If a tree T is a general corona, then T belongs to S3. Proof From Observation 3, the set of external vertices of T is a dominating 2-packing containing all leaves of T and, consequently, by Theorem 7, T ∈ S3. Lemma 9 If a tree T belongs to S3, then T is a general corona. Proof We use induction on n, the number of vertices of a tree. The smallest tree belonging to S3 is a path P4 and, obviously, P4 is a P-corona of P2. Let T ∈ S3 be a tree on n vertices, n > 4. We will show that T is a general corona. Let P = (v0, v1, . . . , vk ) be the longest path in T . From the choice of P, since T does not have a strong support vertex (by Theorem 7), it follows that k 4 and dT (v1) = 2. We consider two cases: dT (v2) = 2, dT (v2) > 2. Case 1: dT (v2) = 2. Let T1 and T2 denote subtrees T [{v0, v1, v2, v3}] and T − {v0, v1, v2}, respectively. By Theorem 7, the tree T has a dominating 2-packing S containing all leaves of T and certainly {v0, v3} ⊆ S. Consequently, S − {v0} is a dominating 2-packing in T2 containing all leaves of T2. Again by Theorem 7, the tree T2 belongs to S3. Thus, by induction, T2 is a general corona. Since v3 belongs to S − {v0}, by Observation 3 and Theorem 7, v3 ∈ E x t (T2). Obviously T1 = P4 is a general corona. Because v3 is also an external vertex in T1, and trees T1 and T2 do not share any other vertex, T = T1 ∪ T2 is a general corona by Observation 4. Case 2: dT (v2) > 2. In this case, again by Theorem 7, the tree T has a dominating 2-packing S containing all leaves of T . Let v be the unique neighbor of v2 belonging to S. Since S is a 2-packing containing all leaves of T , v is not a support vertex in T . Thus, from the choice of P , it follows that either v is a leaf or v = v3. In both cases, let T1 and T2 be subtrees T [{v0, v1, v2, v }] and T − {v0, v1}, respectively. It is easy to observe that S − {v0} is a dominating 2-packing in T2 containing all leaves of T2. Now, again by Theorem 7, the tree T2 belongs to S3. Thus, by induction, T2 is a general corona. Since v belongs to S − {v0}, by Observation 3 and Theorem 7, v ∈ E x t (T2). Certainly T1 = P4 is a general corona and v is external vertex in T1. In addition, T1 and T2 share only the edge v2v . Consequently, by Observation 6, the tree T = T1 ∪ T2 is a general corona. Taking into account Observation 2, Theorem 7, Lemmas 8 and 9 we have the summary result. Theorem 10 Let T be a tree with at least three vertices. Then, the following properties are equivalent: 1. The domination subdivision number of T is equal to 3 (i.e., T ∈ S3). 2. T has a unique dominating 2-packing containing all leaves of T . 3. T belongs to the family F . 4. T is a general corona of a tree. 5. There exists a tree T such that T is obtained from the 2-subdivision S2(T ) by a sequence of internal contractions. 6. There exists a tree T such that T is obtained from the corona T ◦ K1 by a sequence of internal splittings. 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M. Dettlaff, M. Lemańska, J. Topp, P. Żyliński. Coronas and Domination Subdivision Number of a Graph, Bulletin of the Malaysian Mathematical Sciences Society, 2016, 1-8, DOI: 10.1007/s40840-016-0417-0