A New Bound on the Total Domination Subdivision Number

Graphs and Combinatorics (2009) 25:41–47 Digital Object Identifier (DOI) 10.1007/s00373-008-0824-6 Graphs and Combinatorics © Springer-Verlag 2009 A New Bound on the Total Domination Subdivision Number O. Favaron1 , H. Karami2 , R. Khoeilar2,∗ , S. M. Sheikholeslami2,† 1 Univ Paris-Sud, LRI, UMR 8623, Orsay, F-91405, France. e-mail: [email protected] 2 Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, I.R. Iran. e-mail: [email protected] Abstract. A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number γt (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for every simple connected graph G of order n ≥ 3, sdγt (G) ≤ 3 + min{d2 (v); v ∈ V and d(v) ≥ 2} where d2 (v) is the number of vertices of G at distance 2 from v. Key words. Total domination number, Total domination subdivision number. 1. Introduction Let G = (V (G), E(G)) be a simple graph of order n with no isolated vertices. The neighborhood of a vertex u is denoted by N G (u) and its degree |N G (u)| by degG (u). The minimum and maximum degrees of G are respectively denoted by δ(G) and ∆(G) (briefly V, E, N (u), deg(u), δ, ∆ when no ambiguity on the graph is possible). The distance between two vertices u and v is the length of a shortest path joining them. We denote by N2 (v) the set of vertices at distance 2 from the vertex v and put d2 (v) = |N2 (v)| and δ2 (G) = min{d2 (v); v ∈ V (G)}. A set S of vertices of G is a total dominating set (TDS for short) if it is a dominating set of G such that the subgraph G[S] has no isolated vertex. The minimum cardinality of a total dominating set, denoted by γt (G), is called the total domination number of G. A γt (G)-set is a total dominating set of G of cardinality γt (G). When an edge uv of G is subdivided by inserting a new vertex x between u and v, the total domination number does not decrease (Observation 1). The total domination subdivision number sdγt (G) is the minimum number of edges of G that must be subdivided in order to increase the total domination number. This concept was first introduced in [10] for domination, and then extended to total domination in [7]. Since the total ∗ Research supported by the Research Office of Azarbaijan University of Tarbiat Moallem. † Corresponding author. 42 O. Favaron et al. domination number of the graph K 2 does not change when its only edge is subdivided, in the study of the total domination subdivision number we must assume that one of the components of the sgraph has order at least 3. If G 1 , . . . , G s are the γt (G i ) and if G 1 , . . . , G r are the components components of G, then γt (G) = i=1 of G of order at least 3, then sdγt (G) = min {sdγt (G i ) ; 1 ≤ i ≤ r }. Hence, it is sufficient to study sdγt (G) for connected graphs. The parameter sdγt can take large values, although it cannot exceed 3 for trees. The reader can find in [8] a construction showing that there exist connected graphs of arbitrarily large order n satisfying sdγt (G) > log2 n/3. Therefore an interesting problem is to find good upper bounds on sdγt (G) in terms of the order and possibly of other parameters of G. Some bounds are already known. For instance it has been proved that for any graph G of order n, sdγt (G) ≤ n − γt (G) + 1 [5] and sdγt (G) ≤ 2n/3 [6]. Recently, Karami et al. [9] proved that for any simple connected graph G of order n ≥ 3, sdγt (G) ≤ n − δ + 2 (1) with equality if and only if G is isomorphic to K n (n ≥ 4). Our purpose in this paper is to improve this bound by considering δ2 and ∆ instead of δ. As a corollary of the main result we show that sdγt (G) ≤ n − ∆ + 2. We will use the following observations and properties. Observation 1. Let G be simple connected graph G of order n ≥ 3 and S ⊆ E(G). If G ′ is obtained from G by subdividing the edges of S, then γt (G ′ ) ≥ γt (G). Observation 2. Let G ′ be constructed from G by subdividing a set of edges with a set T of subdivising vertices and let D be a γt (G ′ )-set such that D ∩ T = ∅. Since every vertex of D ∩ T is adjacent in G ′ with a vertex of D ∩ V (G), the set D \ T is dominating in G. Hence when we claim that D \ T is a TDS of G, we just have to check that in G this set has no isolated vertex. Theorem 1 [8]. For any connected graph G with adjacent vertices u and v, each of degree at least two, sdγt (G) ≤ d(u) + d(v) − |N (u) ∩ N (v)| − 1 = |N (u) ∪ N (v)| − 1. To simplify the writing, we say that a set A is smaller than a set B when |A| < |B|. We use [11] for terminology and notation which are not defined here. 2. The New Upper Bound We make use of the following lemmas in the proof of Theorem 3. A support vertex is a vertex adjacent with a leaf. Lemma 1. Let G be a simple connected graph. If v ∈ V (G) is a support vertex contained in a triangle, then sdγt (G) ≤ 2. A New Bound on the Total Domination Subdivision Number 43 Proof. Assume N (v) = {v1 , . . . , vdeg(v) }, deg(v1 ) = 1 and v2 v3 ∈ E(G). Let G ′ be obtained from G by subdividing the edges vv1 and v2 v3 , with vertices x, y, respectively. Let D be a γt (G ′ )-set. Without loss of generality we may assume x, v ∈ D. In order to totally dominate y, we must have D ∩ {v2 , v3 } = ∅. Then obviously  D \ {x, y} is a TDS for G smaller than D. Hence, sdγt (G) ≤ 2. Lemma 2. Let G be a simple connected graph of order n ≥ 3. If v ∈ V (G) is a support vertex and has a neighbor u with N (u) \ N [v] = ∅, then sdγt (G) ≤ 2 + |N (u) \ N [v]|. Proof. If N (u) ∩ N (v) = ∅, then the result follows by Lemma 1. Assume N (v) ∩ N (u) = ∅. Let N (v) = {u = v1 , v2 , . . . , vdeg(v) } where deg(v2 ) = 1, and N (u) \ N [v] = {y1 , y2 , . . . , yk }. Let G ′ be obtained from G by subdividing the edge vvi with a vertex u i for i = 1, 2, and the edge uy j with a vertex z j for 1 ≤ i ≤ k. Let Z be the set of the k + 2 subdividing vertices and let D be a γt (G ′ )-set. Without loss of generality we may assume v, u 2 ∈ D. If u ∈ D, then obviously D \ Z is a γt (G)-set. Let u ∈ D. In order to dominate u, we must have D ∩ (Z \ {u 2 }) = ∅. Then (D \ Z ) ∪ {u} is a TDS of G smaller than D. Thus sdγt (G) ≤ 2 + |N (u) \ N [v]| and the proof is complete.  Lemma 3. Let G be a simple connected graph of order n ≥ 3. If G has a vertex v ∈ V (G) which is contained in a triangle vuw such that N (u) ∪ N (w) ⊆ N [v], then sdγt (G) ≤ 3. Proof. Let G ′ be obtained from G by subdividing the edges vu, vw, uw with vertices x, y, z, respectively. Let D be a γt (G ′ )-set. In order to totally dominate x, y and z, the set D must contain at least one endvertex of each edge uv, vw and wu. Hence |D ∩ {u, v, w}| ≥ 2. By Observation 2, if D ∩ {x, y, z} = ∅ then D \ {x, y, z} is a TDS of G smaller than D. Suppose now D ∩ {x, y, z} = ∅. To totally dominate v, D contains a vertex in N G (v) \ {u, w}. If v ∈ D then, since D ∩ {u, w} = ∅, the set D \ {u, w} is a TDS of G smaller than D. If v ∈ / D, then {u, w} ⊆ D and (D \ {u, w}) ∪ {v} is a TDS of G smaller than D. Thus in all cases |D| > γt (G) and the proof is complete.  Lemma 4. Let G be a simple connected graph of order n ≥ 3. If G has a vertex v ∈ V (G) which is contained in a triangle vuw such that N (u) ⊆ N [v] and N (w) \ N [v] = ∅, then sdγt (G) ≤ 3 + |N (w) \ N [v]|. Proof. Let N (w) \ N [v] = {w1 , . . . , wk } and let G ′ be obtained from G by subdividing the edges vu, vw, uw with vertices x, y, z, respectively, and for each i between 1 and k, the edge wwi with the vertex z i . We put T = {z 1 , z 2 , . . . , z k } and also consider the graph G ′′ obtained from G by only inserting the vertices of T . By Observation 1, γt (G ′ ) ≥ γt (G ′′ ) ≥ γt (G). Let D be a γt (G ′ )-set. As in Lemma 3, |D ∩ {u, v, w}| ≥ 2 and if D ∩ {x, y, z} = ∅ then D \ {x, y, z} is a TDS of G ′′ smaller than D. We now suppose that D ∩ {x, y, z} = ∅. 44 O. Favaron et al. If {v, u} ⊆ D then, since v is not isolated in G[D \ {u}] and N G (u) ⊆ N G [v], D \ {u} is a TDS of G ′′ smaller than D. If {v, u}  D, then {v, w} ⊆ D or {u, w} ⊆ D. If D ∩ T = ∅, then D \ T is a TDS of G smaller than D. Suppose now D ∩ T = ∅. Then the set D \ {w} if {v, w} ⊆ D, or (D \ {u, w}) ∪ {v} if {u, w} ⊆ D, is a TDS of G smaller than D.  In all cases we showed that γt (G ′ ) > γt (G), which completes the proof. Lemma 5. Let G be a simple connected graph of order n ≥ 3 and v a vertex of degree at least 2 of G such that (i) N (y) \ N [v] = ∅ for each y ∈ N (v), (ii) there exists a pair α, β of vertices in N (v) such that (N (α) ∩ N (β)) \ N [v] = ∅. Then sdγt (G) ≤ 3 + |N2 (v)|. Proof. Let v1 , v2 be any pair of adjacent vertices of N (v) satisfying (ii) if such a pair exists, otherwise any pair of vertices of N (v) satisfying (ii). Let S = {v1 , v2 , . . . , vk } be a largest subset of N (v) containing v1 , v2 and such that every pair α, β of verk N (v )) \ N [v]. We put N (v ) \ N [v] = tices of S satisfies (ii), and let K = (∪i=1 i i {vi1 , vi2 , . . . , viℓi } for 1 ≤ i ≤ k. Let G ′ be obtained from G by subdividing the edges vv1 and vv2 with respectively z 1 and z 2 , and for each i between 1 and k, the edges vi vi j , 1 ≤ j ≤ ℓi , with a set Ti of ℓi vertices. We put T = ∪1≤i≤k Ti . When v1 and v2 are adjacent, we also subdivide the edge v1 v2 with a vertex u. Let D be a γt (G ′ )-set. In what follows, the expression D \ {u} means D when u does not exists, that is when v1 v2 ∈ / E(G). Case 1. v ∈ / D. Then, to totally dominate z 1 and z 2 , D contains v1 and v2 . Let G ′1 be obtained from G by only inserting the vertices of T3 ∪ · · · ∪ Tk (when k = 2 then G ′1 = G). If D ∩ (T1 ∪ T2 ) ≥ 2, then (D \ (T1 ∪ T2 ∪ {z 1 , z 2 , u})) ∪ {v} is a TDS of G ′1 smaller than D. If |D ∩ (T1 ∪ T2 )| ≤ 1, let without loss of generality D ∩ T1 = ∅. If (T2 ∪ {u, z 1 , z 2 }) ∩ D = ∅, then (D \ (T2 ∪ {z 1 , z 2 , u, v1 })) ∪ {v} is a TDS of G ′1 smaller than D. If (T2 ∪ {u, z 1 , z 2 }) ∩ D = ∅, then v has a neighbor in D \ {v1 , v2 } and (D \ {v1 , v2 }) ∪ {v} is a TDS of G ′1 smaller than D. Case 2. v ∈ D. We consider two subcases. Subcase 2.1. D ∩ (N (v) \ S) = ∅. Let y ∈ D ∩ (N (v) \ S). If D ∩ S = ∅, let z ∈ D ∩ S (the vertex z may be one of v1 , v2 or not). Let Z be the set of vertices subdividing the edges of G incident with z and let G ′2 be the graph obtained from G by inserting all the subdivision vertices used to construct G ′ except those of Z . If D ∩ Z = ∅, then D \ {z} is a TDS of G ′2 smaller than D. If D ∩ Z = ∅, then D \ Z is a TDS of G ′2 smaller than D. A New Bound on the Total Domination Subdivision Number 45 If D ∩ S = ∅ (in this case, v1 v2 ∈ / E(G) and u does not exists), D must contain the set K to totally dominate all the vertices of T in G ′ . Moreover, by the definition of S, every vertex of N (v) \ S has a neighbor in K . Let X = D ∩ T and X ′ = N G ′ (X ) ∩ S. Then |X ′ | ≤ |X | and (D \ (X ∪ {z 1 , z 2 , v})) ∪ X ′ is a TDS of G smaller than D. Subcase 2.2. D ∩ (N (v) \ S) = ∅. If {z 1 , z 2 } ⊆ D, then (D \ {z 1 , z 2 , u}) ∪ {v1 } is a TDS of G ′3 smaller than D, where the graph G ′3 is obtained from G by only inserting the set T of subdivision vertices. If |D ∩ {z 1 , z 2 }| = 1, let z 1 ∈ D and z 2 ∈ D. If D ∩ S = ∅, then D \ {z 1 , u} is a TDS of the graph G ′3 . If D ∩ S = ∅ (in which case u does not exist), then K ⊆ D and every vertex of N (v) \ S has a neighbor in K . As in Subcase 2.1, let X = D ∩ T and X ′ = N G ′ (X ) ∩ S. Then |X ′ | ≤ |X | and (D \ (X ∪ {z 1 , v})) ∪ X ′ ∪ {v1 } is a TDS of G smaller than D. If D ∩ {z 1 , z 2 } = ∅, then v has a neighbor in D ∩ (S \ {v1 , v2 }), say v3 ∈ D. If D ∩ T3 = ∅ then D \ T3 is a TDS of G ′4 smaller than D, where G ′4 is obtained from G by inserting the vertices of (T \ T3 ) ∪ {z 1 , z 2 , u}. We now assume D ∩ T3 = ∅. If D ∩ {v1 , v2 } = ∅, let v1 ∈ D. If D ∩ (T1 ∪ {u}) = ∅, then D \ (T1 ∪ {u}) is a TDS of G ′5 smaller than D, where the graph G ′5 is obtained from G by inserting the vertices of T \ T1 . If D ∩ (T1 ∪ {u}) = ∅, then D \ {v1 } is a TDS of G ′5 . Finally let / E(G), for otherwise u could not be dominated. By D ∩ {v1 , v2 } = ∅. Then v1 v2 ∈ the choice of the pair v1 , v2 , the set S is independent. In order to totally dominate v1 , D ∩ T1 = ∅. Let G ′6 be obtained from G by inserting the subdividing vertices of the set T \ (T1 ∪ T3 ). Since D ∩ T3 = ∅, the set (D \ (T1 ∪ {v3 })) ∪ {v1 } is a TDS of G ′6 smaller than D. In each case, we found a graph G i′ constructed from G by inserting a subset of the subdivision vertices used to construct G ′ and such that γt (G i′ ) < γt (G ′ ). By Observation 1, γt (G ′ ) > γt (G i′ ) ≥ γt (G). Since G ′ was obtained by inserting at most 3 + |T | ≤ 3 + |N2 (v)| vertices, sdγt (G) ≤ 3 + |N2 (v)|.  Lemma 6. Let G be a simple connected graph of order n ≥ 3 and v a vertex of degree at least 2 of G such that (i) N (y) \ N [v] = ∅ for each y ∈ N (v), (ii) (N (α) ∩ N (β)) \ N [v] = ∅ for every pair α, β of vertices in N (v). Then sdγt (G) ≤ 3 + |N2 (v)|. Proof. Let N (v) = {v1 , . . . , vk } and M = N (v1 ) \ N [v] = {w1 , . . . , w p }. It follows from the hypothesis that each y ∈ N (v)\{v1 } has a neighbor in M. Let T be a largest subset of N (v)\{v1 } such that for each subset T1 ⊆ T , |N (T1 )\(N [v]∪ M)| ≥ |T1 |. By the definition of T , |N2 (v)| ≥ |M|+|T | and for every vertex u in U = N (v)\(T ∪{v1 }), N (u) \ N [v] ⊆ M ∪ N (T ). Moreover M dominates N (v) by (ii). 46 O. Favaron et al. Case 1. |U | ≤ 1. By Theorem 1, sdγt (G) ≤ |N (v) ∪ N (v1 )| − 1 = |T | + |U | + 1 + |M| + 1 − 1 ≤ |U | + 1 + |N2 (v)| ≤ |N2 (v)| + 2. Case 2. |U | ≥ 2. Let T = ∅ or, without loss of generality, T = {v2 , . . . , vs }. Let G ′ be obtained from G by subdividing the |M| + |T | + 3 edges v1 w j for 1 ≤ j ≤ k and vvi for 1 ≤ i ≤ s + 2 (1 ≤ i ≤ 3 if T = ∅). Let Z 1 be the set of the vertices subdividing the edges v1 w j for 1 ≤ j ≤ k, Z 2 the set of the vertices subdividing the edges vvi for 1 ≤ i ≤ s + 2 and Z = Z 1 ∪ Z 2 . Let D be a γt (G ′ )-set. First assume that v ∈ D. Then {v1 , . . . , vs+2 } ⊆ D to totally dominate the vertices of Z 2 and since N ({vs+1 , vs+2 })\ N [v] ⊆ M ∪ N (T ), (D\(Z ∪{vs+1 , vs+2 }))∪ {v} is a TDS of G smaller than D. Now assume that v ∈ D and v1 ∈ / D. Then M ⊆ D. If D ∩ Z 2 = ∅, then (D \ (Z 2 ∪ Z 1 ∪ {v})) ∪ {v1 } is a TDS of G smaller than D. If D ∩ Z 2 = ∅, then D contains a vertex v ′ ∈ U \ {vs+1 , vs+2 } (note that v ′ has a neighbor in M). If D ∩ Z 1 = ∅, then (D \ (Z 1 ∪{v})) ∪{v1 } is a TDS of G smaller than D. If D ∩ Z 1 = ∅, then D \ {v} is a TDS of G smaller than D. Finally assume that v ∈ D and v1 ∈ D. If D ∩ Z = ∅, then D \ Z is a TDS of G smaller than D. If D ∩ Z = ∅, then D contains a neighbor of v belonging to U \ {vs+1 , vs+2 } and D \ {v1 } is a TDS of G smaller than D. Therefore γt (G) < γt (G ′ ) and sdγt (G) ≤ |M| + |T | + 3 ≤ |N2 (v)| + 3.  We are now ready to prove our main result. Theorem 3. Let G be a simple connected graph of order n ≥ 3. Then sdγt (G) ≤ 3 + min{d2 (v) ; v ∈ V and d(v) ≥ 2}. Proof. If G is a star K 1,n−1 then sdγt (G) = 2. Otherwise, let v be a vertex of degree at least 2 of G such that d2 (v) is minimum. The result is a consequence of Lemmas 1 and 2 if v is a support vertex, of Lemmas 3 and 4 if some neighbor u of v different from a leaf satisfies N (u) ⊆ N [v], and of Lemmas 5 and 6 if N (y) ∩ N [v] = ∅ for every y ∈ N (v).  Corollary 1. Let G be a connected graph of minimum degree at least 2. Then sd γt (G) ≤ δ2 (G) + 3. For a vertex v of degree ∆, |N2 (v)| ≤ n − ∆ − 1. Therefore the following improvement of the bound (1) is an immediate corollary of Theorem 3. Corollary 2. Let G be a connected graph of order n ≥ 3. Then sdγt (G) ≤ n − ∆ + 2. A New Bound on the Total Domination Subdivision Number 47 The complete graphs K n (n ≥ 4), which form the extremal class for (1), are obviously also extremal for Theorem 3 and Corollary 2. Another example of graphs which are extremal for Theorem 3, Corollary 2 and Corollary 1, although not for (1), is given by the complete split graphs K m ∨ K n−m for 1 ≤ m ≤ n − 3. As shown in [9], these graphs form the class of graphs such that γt (G) = 2 and sdγt (G) = 3. One can wonder whether the new bound is also an improvement for the two other bounds cited in the introduction. When G is a cycle Cn with n = 12k, then 2n/3 = 8k, n − γt (G) = 6k and δ2 (G) + 3 = 5. Therefore for cycles, the bound of Theorem 3 is much better than the other ones. This is not always the case. Erdős and Rényi [3] constructed with the the aid of projective planes √ an infinite family of graphs with diameter 2 and ∆ asymptotically equal to n. For these graphs, δ2 + 3 = n − ∆ + 2 > 2n/3. To conclude the paper, let us mention the following conjecture proposed in [4] and established in some classes of graphs. Conjecture 1. For any connected graph of order n ≥ 3, sdγt (G) ≤ γt (G) + 1. 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