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Upper bounds on the upper signed total domination number of graphs
Discrete Applied Mathematics 157 (2009) 1098–1103
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Upper bounds on the upper signed total domination number of graphsI Erfang Shan a,b,∗ , T.C.E. Cheng b a Department of Mathematics, Shanghai University, Shanghai 200444, PR China b Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
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Article history: Received 20 October 2007 Received in revised form 3 April 2008 Accepted 4 April 2008 Available online 20 May 2008 Keywords: Dominating function Upper signed total domination Upper bound
a b s t r a c t Let G = (V, E) be a graph. A function f : V → {−1, +1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function g, f 6= g, for which g(v) ≤ f (v) for every v ∈ V . The weight of a signed total dominating function is the sum of its function values over all vertices of G. The upper signed total domination number of G is the maximum weight of a minimal signed total dominating function on G. In this paper we present a sharp upper bound on the upper signed total domination number of an arbitrary graph. This result generalizes previous results for regular graphs and nearly regular graphs. © 2008 Elsevier B.V. All rights reserved.
1. Introduction In this paper we in general follow [5] for notation and graph theory terminology. Specifically, let G = (V, E) with vertex set V and edge set E, and let v be a vertex in V . The open neighborhood of v, N(v), is defined as the set of vertices adjacent to v, i.e., N(v) = {u | uv ∈ E}. The closed neighborhood of v is N[v] = N(v) ∪ {v}. For a subset S ⊆ V , the open neighborhood of S S S is N(S) = v∈S N(v) and the closed neighborhood of S is N[S] = v∈S N[v]. We write deg(v) for the degree of v in G, which equals |N(v)|. If each vertex in G has an odd degree, then we call G an odd-degree graph. For S ⊆ V , we let degS (v) denote the number of vertices in S that are adjacent to v. In particular, degV (v) = deg(v). If deg(v) = k for all v ∈ V , then we call G a k-regular graph. If deg(v) = k − 1 or k for all v ∈ V , then we call G a nearly k-regular graph. The subgraph of G induced by S is denoted by hSi. If A, B ⊆ V , A ∩ B = ∅, then let e(A, B) be the number of edges between A and B. The maximum degree among the vertices of G is denoted by ∆(G) and the minimum degree by δ(G). P P For a real-value function f : V → R, the weight of f is w(f ) = v∈V f (v), and for S ⊆ V we define f (S) = v∈S f (v), so w(f ) = f (V ). For a vertex v in V , we denote f (N(v)) by f [v] for notational convenience. A signed total dominating function of a graph G is defined in [14] as a function f : V → {+1, −1} such that for every vertex P v, f [v] ≥ 1, and the minimum cardinality of the weight w(f ) = v∈V f (v) over all such functions is called the signed total s domination number of G, denoted by γt (G), i.e.,
γts (G) = min{f (V ) : f is a signed dominating function of G}. We say that f is a minimal signed total dominating function if there does not exist a signed total dominating function g : V → {+1, −1}, f 6= g, for which g(v) ≤ f (v) for every vertex v ∈ V . The upper signed total domination number of G,
I Research was partially supported by the Hong Kong Polytechnic University under grant number G-YX69, the National Natural Science Foundation of China under grant numbers 10571117, 60773078 and the ShuGuang Plan of Shanghai Education Development Foundation under grant number 06SG42. ∗ Corresponding author at: Department of Mathematics, Shanghai University, Shanghai 200444, PR China. Tel.: +86 21 66134843. E-mail address: [email protected] (E. Shan).
0166-218X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2008.04.005
E. Shan, T.C.E. Cheng / Discrete Applied Mathematics 157 (2009) 1098–1103
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denoted by Γts (G), is the maximum weight of a minimal signed total dominating function of G. For a signed total dominating function of weight Γts (G), we call it a Γts (G)-function. The concept of signed total domination in graphs was introduced by Zelinka [14] and studied in [7,8,12]. The signed total domination can be viewed as a proper generalization of the classical total domination [1] and signed domination [2,3] in graph theory. Henning [7] observed the fact that the signed total domination differs significantly from signed domination. By simply changing {+1, −1} in the definition of signed total domination to {+1, 0, −1}, Harris and Hattingh [4] put forward the concept of minus total domination of graphs. Minus total domination has been studied in, for example, [10,11,13]. A survey on this topic of dominating functions in graph theory can be found in [6,9]. The decision problem for the signed total domination number of a graph has been shown to be NP-complete, even when the graph is restricted to a bipartite graph or a chordal graph [7]. It is therefore of interest to determine bounds on the domination parameter of a graph. Henning [7] established a sharp lower bound on γts of a general graph in terms of its minimum degree, maximum degree and order. In this paper we study the upper signed domination number Γts of a graph. We shall present a sharp upper bound on Γts of an arbitrary graph. Henning [7] gave sharp upper bounds on Γts of a regular graph in terms of its order. Theorem 1 (Henning [7]). If G = (V, E) is a k-regular graph with k ≥ 1 of order n, then ! k2 + 1 n for k odd, k2 + 2k − 1 s ! Γt (G) ≤ k2 + k + 2 n for k even. 2 k + 3k − 2 Furthermore, these bounds are sharp. Further, Kang and Shan [8] presented sharp upper bounds on Γts of a nearly regular graph in terms of its order. Theorem 2 (Kang and Shan [8]). If G = (V, E) is a nearly (k + 1)-regular graph with k ≥ 1 of order n, then ! 2 k + 3k + 4 n for k odd, k2 + 5k + 2 ! Γts (G) ≤ (k + 1)2 + 3 n for k even. k(k + 4) Furthermore, these bounds are sharp. One may ask what the upper bounds on Γts for an arbitrary graph are. Our purpose here is to present a sharp upper bound on Γts of an arbitrary graph in terms of its minimum degree, maximum degree and order. The main result of this paper is the following. Theorem 3. If G = (V, E) is a graph of order n with minimum degree δ and maximum degree ∆, then ∆(δ + 3) − (δ − 1) n Γts (G) ≤ ∆(δ + 3) + (δ − 1) ∆(δ + 2) − (δ − 2) n ∆(δ + 2) + (δ − 2)
for δ odd, for δ even.
Moreover, if G is an odd-degree graph, then Γts (G) ≤
∆(δ + 1) − (δ − 1) ∆(δ + 1) + (δ − 1)
n.
Furthermore, these bounds are sharp. Clearly, if ∆ = δ = k, then (∆(δ + 1) − (δ − 1))n/(∆(δ + 1) + (δ − 1)) = (k2 + 1)n/(k2 + 2k − 1) for k being odd, and (∆(δ + 2) − (δ − 2))n/(∆(δ + 2) + (δ − 2)) = (k2 + k + 2)n/(k2 + 3k − 2) for k being even. If ∆ = k + 1 and δ = k, then (∆(δ + 3) − (δ − 1))n/(∆(δ + 3) + (δ − 1)) = (k2 + 3k + 4)/(k2 + 5k + 2) for k being odd, and (∆(δ + 2) − (δ − 2))n/(∆(δ + 2) + (δ − 2)) = ((k + 1)2 + 3)/k(k + 4) for k being even. Thus, Theorems 1 and 2 are special cases of Theorem 3.
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2. Proof of Theorem 3 To prove Theorem 3, we shall need the following lemmas due to Henning [7]. Observing that the complete graph Kn is 1-factorable for n being even, we immediately have Lemma 1 (Henning [7]). If k and n are integers with k < n and n is even, then we can construct a k-regular graph on n vertices. Lemma 2 (Henning [7]). A signed total dominating function of a graph G = (V, E) is minimal if and only if for every vertex v ∈ V with f (v) = 1, there exists a vertex u ∈ N(v) with f [u] ≤ 2, where if deg(u) is odd, then f [u] = 1; if deg(u) is even, then f [u] = 2. Now we can present the proof of Theorem 3. Proof of Theorem 3. Let f be a Γts (G)-function of G, and let P and M denote the sets of those vertices in G that are assigned under f the value +1 and −1, respectively. If δ = 1 or 2, then the results are trivial. Hence in what follows we assume that δ ≥ 3. For notational convenience, we write b(δ − 1)/2c = s1 , d(δ + 1)/2e = s2 , b(∆ − 1)/2c = t1 and d(∆ + 1)/2e = t2 . Obviously, f [v] = deg(v) − 2degM (v) for any vertex v ∈ V . Then, by Lemma 2, M 6= ∅. Let |P| = p and |M| = m. Thus, w(f ) = |P | − |M| = n − 2m. For any vertex v ∈ V , degM (v) ≤ b(deg(v) − 1)/2c for otherwise f [v] < 1. Hence we can partition P into t1 + 1 sets by defining Pi = {v ∈ P | degM (v) = i} and letting |Pi | = pi for i = 0, 1, . . . , t1 . Then we have n=m+p=m+
t1 X
pi .
(1)
i =0
For any vertex v ∈ V , degP (v) ≥ d(deg(v) + 1)/2e for otherwise f [v] < 1. We define Mj = {v ∈ M | degP (v) = j}
for j = s2 , s2 + 1, . . . , t2 Pt2 0 and M0 = M − j=s2 Mj . Let |Mj | = mj , and so |M0 | = m − j= s2 mj . Clearly, (Ms2 , . . . , Mt2 , M ) is a partition of M. Since each 0 vertex in M is adjacent to at most ∆ vertices of P, we have St2
t1 X
ipi = e(P, M) ≤ (s2 ms2 + · · · + t2 mt2 ) + ∆ m − (ms2 + · · · + mt2 )
.
i =1
Hence, t1 X
ipi ≤ ∆m −
(∆ − s2 )ms2 + · · · + (∆ − t2 )mt2 .
(2)
i =1
If P0 = ∅, then, by (1) and (2), we have n=m+
t1 X i=1
pi ≤ m +
t1 X
ipi ≤ (∆ + 1)m.
i =1
This implies that m ≥ n/(∆ + 1), and so Γts (G) = n − 2m ≤ (∆ − 1)n/(∆ + 1). Observing that ∆(δ + 3) − (δ − 1) ∆(δ + 2) − (δ − 2) (∆ − 1)n/(∆ + 1) ≤ min n, n , ∆(δ + 3) + (δ − 1) ∆(δ + 2) + (δ − 2) then the desired result follows. We therefore may assume that P0 6= ∅. Ss1 −1 0 According to our partition for P and M, we have f [v] ≥ 3 for any v ∈ ( i= 0 Pi ) ∪ M , so if f [v] = 1 or 2 for v ∈ V , then St1 St2 v ∈ ( i=s1 Pi ) ∪ ( j=s2 Mj ). For any vertex v ∈ P0 , since f is minimal, by Lemma 2, there exists a vertex u ∈ N(v) such that St1 f [u] ∈ {1, 2}. Let I = {u ∈ N(P0 ) | f [u] = 1 or 2}. Then I ⊆ i= s1 Pi . So p0 = |P0 | ≤ e(P0 , I) = e P0 ,
t1 [
!
(Pi ∩ I) ≤ e P0 ,
i=s1
t1 [
! Pi
.
(3)
i=s1
Furthermore, for every vertex u ∈ Pi ∩ I (s1 ≤ i ≤ t1 ), there must exist a neighbor u0 of u such that f [u0 ] = 1 or 2. Note that u0 St1 St2 St1 St2 0 0 belongs to ( i= s1 Pi ) ∪ ( j=s2 Mj ). If u ∈ i=s1 Pi , then u is adjacent to at most i + 1 vertices of P0 , while if u ∈ j=s2 Mj , then u is adjacent to at most i + 2 vertices of P0 . Hence we can write Pi ∩ I (s1 ≤ i ≤ t1 ) as the disjoint union of two sets IPi0 and IPi00 , where IPi0 = {u ∈ Pi ∩ I | degP0 (u) = i + 2} and IPi00 = Pi ∩ I − IPi0 . Then IPi00 = {u ∈ Pi ∩ I | degP0 (u) ≤ i + 1}. Let |IPi0 | = p0i , and St1 St2 0 so |IPi00 | = |Pi ∩ I| − p0i . Since each vertex u ∈ i= s1 IPi is adjacent to a vertex in j=s2 Mj , it follows that t1 X i=s1
p0i ≤ s2 ms2 + (s2 + 1)m(s2 +1) + · · · + t2 mt2 .
(4)
E. Shan, T.C.E. Cheng / Discrete Applied Mathematics 157 (2009) 1098–1103
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Thus, by Eqs. (3) and (4), we have p0 ≤
t1 X
(i + 2)p0i +
t1 X
(i + 2)p0i +
t1 X
t1 X
(i + 1)(pi − p0i )
i=s1
i=s1
≤
(i + 1)(|Pi ∩ I| − p0i )
i=s1
i=s1
≤
t1 X
(i + 1)pi +
t2 X
jmj .
(5)
j=s2
i=s1
We now distinguish two possibilities depending on the parity of the minimum degree δ. Case 1. δ is odd. Then s1 = (δ − 1)/2. Noting that (δ + 3)i/(δ − 1) ≥ i + 2 holds when i ≥ (δ − 1)/2, then, by Eqs. (1), (2) and (5), we obtain ! t2 t1 t1 X X X jmj + (i + 1)pi + pi n ≤ m+ i=s1
= m+
t1 X i=s1
(i + 2)pi +
i =1
j=s2 sX 1 −1
pi +
i =1
t2 X
jmj
j=s2
≤ m+
t1 sX t2 1 −1 X δ+3 X ipi + pi + jmj δ − 1 i=s1 j=s2 i=1
≤ m+
t1 t2 X δ+3 X ipi + jmj δ − 1 i =1 j=s2
≤ m+
t2 t2 X δ+3 X δ+3 jmj m∆ − (∆ − j)mj + δ−1 δ − 1 j=s2 j=s2
= m+
t2 X (δ + 3)∆ − 2j(δ + 1) δ+3 m∆ − mj . δ−1 δ−1 j=s2
Further, note that
(δ + 3)∆ − 2j(δ + 1) (δ + 3)∆ − 2t2 (δ + 1) ≥ ≥ 0, δ−1 δ−1 the sign in the above second inequality can be justified by observing the fact that t2 = (∆ + 1)/2 if ∆ = δ and t2 ≤ (∆ + 2)/2 if ∆ ≥ δ + 1. Then n ≤ m + ((δ + 3)∆/(δ − 1))m, which implies that m ≥ (δ − 1)n/(∆(δ + 3) + (δ − 1)). Hence, ∆(δ + 3) − (δ − 1) Γts (G) = w(f ) = n − 2m ≤ n. ∆(δ + 3) + (δ − 1) That the bound is sharp may be seen as follows. For any integers l ≥ 1 and r ≥ l + 1, let Fl,r be the graph with vertex set X ∪ Y ∪ Z with |X | = l, |Y | = 2r and |Z | = 2r(l + 1), where X is an independent set of vertices. The edge set of Fl,r is constructed as follows: Add 2rl edges between X and Y so that hX ∪ Y i forms a complete bipartite graph with partition sets X and Y . Add 2r(l + 1) edges between Y and Z so that each vertex of Y is precisely adjacent to l + 1 vertices of Z and each vertex of Z is precisely adjacent to one vertex of Y . Add edges joining vertices of Y so that Y induces a 1-regular graph. Add edges joining vertices of Z so that Z induces a 2l-regular graph (since 2l < 2r(l + 1) = |Z | and |Z | is even, it follows from Lemma 1 that such an addition of edges is possible). By construction, Fl,r is a graph of order n = l + 2r + 2r(l + 1) with minimum degree δ = 2l + 1 and maximum degree ∆ = 2r. The function g that assigns to each vertex of X the value −1 and to each vertex of Y ∪ Z the value +1 is a minimal signed total dominating function of Fl,r as g[u] = 2 for any u ∈ Y . It is easy to see that ∆(δ + 3) − (δ − 1) w(g) = 2r + 2r(l + 1) − l = n. ∆(δ + 3) + (δ − 1) Consequently, Γts (Fl,r ) = (∆(δ + 3) − (δ − 1))n/(∆(δ + 3) + (δ − 1)). Now suppose that G is an odd-degree graph. We shall improve the above bound. Since all vertices in G have odd degree, it follows that for any vertex v ∈ V , f [v] = deg(v) − 2degM (v) is odd. Thus, for every vertex u ∈ Pi ∩ I (s1 ≤ i ≤ t1 ), by Lemma 2, St1 St2 0 there exist u0 ∈ N(u) such that f [u0 ] = 1. If u0 ∈ i= s1 Pi , then u is adjacent to at most i vertices of P0 , while if u ∈ j=s2 Mj , then u is adjacent to at most i + 1 vertices of P0 . Hence we can partition Pi ∩ I (s1 ≤ i ≤ t1 ) into two sets IPi0 and IPi00 , where IPi0 = {u ∈ Pi ∩ I | degP0 (u) = i + 1} and IPi00 = {u ∈ Pi ∩ I | degP0 (u) ≤ i}. Let |IPi0 | = p0i , and so |IPi00 | = |Pi ∩ I| − p0i . Therefore, by
E. Shan, T.C.E. Cheng / Discrete Applied Mathematics 157 (2009) 1098–1103
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(3) and (4), we can obtain p0 ≤
t1 X
(i + 1)p0i +
t1 X
i(|Pi ∩ I| − p0i ) ≤
ipi +
i=s1
i=s1
i=s1
t1 X
t2 X
jmj .
(6)
j=s2
Since (δ + 1)/(δ − 1)i ≥ i + 1 when i ≥ (δ − 1)/2, it follows from (1), (2) and (6) that ! t2 t1 t1 X X X δ+1 jmj + pi ≤ 1 + ipi + n≤m+ ∆ m. δ−1 j=s2 i=s1 i =1 Consequently, Γts (G) = n − 2m ≤
∆(δ + 1) − (δ − 1)
∆(δ + 1) + (δ − 1)
n.
That the bound is sharp may be seen as follows. For any integers l ≥ 1, r ≥ l and an even integer q, where 2l ≤ q ≤ 2r, let Hl,r be the graph with vertex set X ∪ Y ∪ Z with |X | = 2l2 , |Y | = 2l(2r + 1) and |Z | = 2l2 (2r + 1), where X is an independent set of vertices. The edge set of Hl,r is constructed as follows: Add 2l2 (2r + 1) edges between X and Y so that each vertex in X has degree 2r + 1 while each vertex in Y has degree l. Add 2l2 (2r + 1) edges between Y and Z so that each vertex of Y is precisely adjacent to l vertices of Z and each vertex of Z is precisely adjacent to one vertex of Y . Add l(2r + 1) edges joining vertices of Y so that Y induces a 1-regular graph. Add edges joining vertices of Z so that Z induces a q-regular graph (since q ≤ 2r < |Z | and |Z | is even, it follows from Lemma 1 that such an addition of edges is possible). By construction, Hl,r is a graph of order n = 2l2 + 2l(2r + 1) + 2l2 (2r + 1) with minimum degree δ = 2l + 1 and maximum degree ∆ = 2r + 1. The function g that assigns to each vertex of X the value −1 and to each vertex of Y ∪ Z the value +1 is a minimal signed total dominating function of Hl,r as g[u] = 1 for any u ∈ Y . It is easy to see that ∆(δ + 1) − (δ − 1) w(g) = 2l(2r + 1) + 2l2 (2r + 1) − 2l2 = n. ∆(δ + 1) + (δ − 1) Consequently, Γts (Hl,r ) = (∆(δ + 1) − (δ − 1))n/(∆(δ + 1) + (δ − 1)). Case 2. δ is even. Then s1 = (δ − 2)/2. Noting that (δ + 2)i/(δ − 2) ≥ i + 2 holds when i ≥ (δ − 2)/2, then, by Eqs. (1), (2) and (5) again, we have ! t1 t2 t1 X X X n ≤ m+ (i + 1)pi + jmj + pi i=s1
= m+
t1 X
(i + 2)pi +
i=s1
i =1
j=s2 sX 1 −1
pi +
i =1
t2 X
jmj
j=s2
≤ m+
t2 sX t1 1 −1 X δ+2 X jmj pi + ipi + δ − 2 i=s1 j=s2 i =1
≤ m+
t1 t2 X δ+2 X ipi + jmj δ − 2 i =1 j=s2
≤ m+
t2 t2 X δ+2 δ+2 X m∆ − (∆ − j)mj + jmj δ−2 δ − 2 j=s2 j=s2
= m+
t2 X δ+2 (δ + 2)∆ − 2jδ m∆ − mj . δ−2 δ−2 j=s2
Further, observing that
(δ + 2)∆ − 2jδ (δ + 2)∆ − 2t2 δ ≥ ≥ 0, δ−2 δ−2 we have n ≤ m + ((δ + 2)∆/(δ − 2))m, which implies that m ≥ (δ − 2)n/(∆(δ + 2) + (δ − 2)). Hence, ∆(δ + 2) − (δ − 2) Γts (G) = w(f ) = n − 2m ≤ n. ∆(δ + 2) + (δ − 2) That the bound is sharp may be seen as follows. For any integers l ≥ 1, r ≥ l + 1 and q, where 2l + 1 ≤ q ≤ 2r − 1, let Gl,r be the graph with vertex set X ∪ Y ∪ Z with |X | = l, |Y | = 2r and |Z | = 2r(l + 1), where X is an independent set of vertices. The edge set of Gl,r is constructed as follows: Add 2rl edges between X and Y so that hX ∪ Y i forms a complete bipartite graph with partition sets X and Y . Add 2r(l + 1) edges between Y and Z so that each vertex of Y is precisely adjacent to l + 1 vertices
E. Shan, T.C.E. Cheng / Discrete Applied Mathematics 157 (2009) 1098–1103
1103
Fig. 1. The graph Gl,r .
of Z and each vertex of Z is precisely adjacent to one vertex of Y . Add edges joining vertices of Y so that Y induces a 1-regular graph. Add edges joining vertices of Z so that Z induces a q-regular graph (since q < 2r(l + 1) = |Z | and |Z | is even, it follows from Lemma 1 that such an addition of edges is possible). The graph Gl,r is shown in Fig. 1. By construction, Gl,r is a graph of order n = l + 2r + 2r(l + 1) with minimum degree δ = 2(l + 1) and maximum degree ∆ = 2r. The function g that assigns to each vertex of X the value −1 and to each vertex of Y ∪ Z the value +1 is a minimal signed total dominating function of Gl,r as g[u] = 2 for any u ∈ Y . It is easy to see that ∆(δ + 2) − (δ − 2) w(g) = 2r + 2r(l + 1) − l = n. ∆(δ + 2) + (δ − 2) Consequently, Γts (Gl,r ) = (∆(δ + 2) − (δ − 2))n/(∆(δ + 2) + (δ − 2)).
Acknowledgements The authors would like to thank the referees for their valuable comments. References [1] E.J. Cockayne, R.M. Dawes, S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211–219. [2] J. Dunbar, S.T. Hedetniemi, M.A. Henning, P.J. Slater, Signed domination in graphs, in: Graph Theory, Combinatorics, and Applications, vol. 1, Wiley, New York, 1995, pp. 311–322. [3] Z. Füredi, D. Mubayi, Signed domination in regular graphs and set-systems, J. Combin. Theory Ser. B 76 (1999) 223–239. [4] L. Harris, J.H. Hattingh, The algorithm complexity of certain functional variations of total domination in graphs, Australas. J. Combin. 29 (2004) 143–156. [5] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. [6] M.A. Henning, Dominating function in graphs, in: Domination in Graphs: Advanced Topics, vol. II, Marcel Dekker Inc., New York, 1998, pp. 31–62. [7] M.A. Henning, Signed total domination in graphs, Discrete Math. 278 (2004) 109–125. [8] L.Y. Kang, E.F. Shan, Signed total domination in nearly regular graphs, J. Shanghai Univ. 10 (2006) 4–8. [9] L.Y. Kang, E.F. Shan, Dominating functions with integer values in graphs—a survey, J. Shanghai Univ. 11 (2007) 437–448. [10] L.Y. Kang, E.F. Shan, L. Caccetta, Total minus domination in k-partite graphs, Discrete Math. 306 (2006) 1771–1775. [11] H.C. Wang, E.F. Shan, Upper minus total domination of a 5-regular graph, Ars Combin. (in press). [12] H.M. Xing, L. Sun, X.G. Chen, Signed total domination in graphs, J. Beijing Inst. Technol. 12 (2003) 319–321. [13] H. Yan, X.Q. Yang, E.F. Shan, Upper minus total domination in small-degree regular graphs, Discrete Math. 307 (2007) 2453–2463. [14] B. Zelinka, Signed total domination number of a graph, Czechoslovak Math. J. 51 (2001) 225–229.
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Upper bounds on the upper signed total domination number of graphsI Erfang Shan a,b,∗ , T.C.E. Cheng b a Department of Mathematics, Shanghai University, Shanghai 200444, PR China b Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
article
info
Article history: Received 20 October 2007 Received in revised form 3 April 2008 Accepted 4 April 2008 Available online 20 May 2008 Keywords: Dominating function Upper signed total domination Upper bound
a b s t r a c t Let G = (V, E) be a graph. A function f : V → {−1, +1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function g, f 6= g, for which g(v) ≤ f (v) for every v ∈ V . The weight of a signed total dominating function is the sum of its function values over all vertices of G. The upper signed total domination number of G is the maximum weight of a minimal signed total dominating function on G. In this paper we present a sharp upper bound on the upper signed total domination number of an arbitrary graph. This result generalizes previous results for regular graphs and nearly regular graphs. © 2008 Elsevier B.V. All rights reserved.
1. Introduction In this paper we in general follow [5] for notation and graph theory terminology. Specifically, let G = (V, E) with vertex set V and edge set E, and let v be a vertex in V . The open neighborhood of v, N(v), is defined as the set of vertices adjacent to v, i.e., N(v) = {u | uv ∈ E}. The closed neighborhood of v is N[v] = N(v) ∪ {v}. For a subset S ⊆ V , the open neighborhood of S S S is N(S) = v∈S N(v) and the closed neighborhood of S is N[S] = v∈S N[v]. We write deg(v) for the degree of v in G, which equals |N(v)|. If each vertex in G has an odd degree, then we call G an odd-degree graph. For S ⊆ V , we let degS (v) denote the number of vertices in S that are adjacent to v. In particular, degV (v) = deg(v). If deg(v) = k for all v ∈ V , then we call G a k-regular graph. If deg(v) = k − 1 or k for all v ∈ V , then we call G a nearly k-regular graph. The subgraph of G induced by S is denoted by hSi. If A, B ⊆ V , A ∩ B = ∅, then let e(A, B) be the number of edges between A and B. The maximum degree among the vertices of G is denoted by ∆(G) and the minimum degree by δ(G). P P For a real-value function f : V → R, the weight of f is w(f ) = v∈V f (v), and for S ⊆ V we define f (S) = v∈S f (v), so w(f ) = f (V ). For a vertex v in V , we denote f (N(v)) by f [v] for notational convenience. A signed total dominating function of a graph G is defined in [14] as a function f : V → {+1, −1} such that for every vertex P v, f [v] ≥ 1, and the minimum cardinality of the weight w(f ) = v∈V f (v) over all such functions is called the signed total s domination number of G, denoted by γt (G), i.e.,
γts (G) = min{f (V ) : f is a signed dominating function of G}. We say that f is a minimal signed total dominating function if there does not exist a signed total dominating function g : V → {+1, −1}, f 6= g, for which g(v) ≤ f (v) for every vertex v ∈ V . The upper signed total domination number of G,
I Research was partially supported by the Hong Kong Polytechnic University under grant number G-YX69, the National Natural Science Foundation of China under grant numbers 10571117, 60773078 and the ShuGuang Plan of Shanghai Education Development Foundation under grant number 06SG42. ∗ Corresponding author at: Department of Mathematics, Shanghai University, Shanghai 200444, PR China. Tel.: +86 21 66134843. E-mail address: [email protected] (E. Shan).
0166-218X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2008.04.005
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denoted by Γts (G), is the maximum weight of a minimal signed total dominating function of G. For a signed total dominating function of weight Γts (G), we call it a Γts (G)-function. The concept of signed total domination in graphs was introduced by Zelinka [14] and studied in [7,8,12]. The signed total domination can be viewed as a proper generalization of the classical total domination [1] and signed domination [2,3] in graph theory. Henning [7] observed the fact that the signed total domination differs significantly from signed domination. By simply changing {+1, −1} in the definition of signed total domination to {+1, 0, −1}, Harris and Hattingh [4] put forward the concept of minus total domination of graphs. Minus total domination has been studied in, for example, [10,11,13]. A survey on this topic of dominating functions in graph theory can be found in [6,9]. The decision problem for the signed total domination number of a graph has been shown to be NP-complete, even when the graph is restricted to a bipartite graph or a chordal graph [7]. It is therefore of interest to determine bounds on the domination parameter of a graph. Henning [7] established a sharp lower bound on γts of a general graph in terms of its minimum degree, maximum degree and order. In this paper we study the upper signed domination number Γts of a graph. We shall present a sharp upper bound on Γts of an arbitrary graph. Henning [7] gave sharp upper bounds on Γts of a regular graph in terms of its order. Theorem 1 (Henning [7]). If G = (V, E) is a k-regular graph with k ≥ 1 of order n, then ! k2 + 1 n for k odd, k2 + 2k − 1 s ! Γt (G) ≤ k2 + k + 2 n for k even. 2 k + 3k − 2 Furthermore, these bounds are sharp. Further, Kang and Shan [8] presented sharp upper bounds on Γts of a nearly regular graph in terms of its order. Theorem 2 (Kang and Shan [8]). If G = (V, E) is a nearly (k + 1)-regular graph with k ≥ 1 of order n, then ! 2 k + 3k + 4 n for k odd, k2 + 5k + 2 ! Γts (G) ≤ (k + 1)2 + 3 n for k even. k(k + 4) Furthermore, these bounds are sharp. One may ask what the upper bounds on Γts for an arbitrary graph are. Our purpose here is to present a sharp upper bound on Γts of an arbitrary graph in terms of its minimum degree, maximum degree and order. The main result of this paper is the following. Theorem 3. If G = (V, E) is a graph of order n with minimum degree δ and maximum degree ∆, then ∆(δ + 3) − (δ − 1) n Γts (G) ≤ ∆(δ + 3) + (δ − 1) ∆(δ + 2) − (δ − 2) n ∆(δ + 2) + (δ − 2)
for δ odd, for δ even.
Moreover, if G is an odd-degree graph, then Γts (G) ≤
∆(δ + 1) − (δ − 1) ∆(δ + 1) + (δ − 1)
n.
Furthermore, these bounds are sharp. Clearly, if ∆ = δ = k, then (∆(δ + 1) − (δ − 1))n/(∆(δ + 1) + (δ − 1)) = (k2 + 1)n/(k2 + 2k − 1) for k being odd, and (∆(δ + 2) − (δ − 2))n/(∆(δ + 2) + (δ − 2)) = (k2 + k + 2)n/(k2 + 3k − 2) for k being even. If ∆ = k + 1 and δ = k, then (∆(δ + 3) − (δ − 1))n/(∆(δ + 3) + (δ − 1)) = (k2 + 3k + 4)/(k2 + 5k + 2) for k being odd, and (∆(δ + 2) − (δ − 2))n/(∆(δ + 2) + (δ − 2)) = ((k + 1)2 + 3)/k(k + 4) for k being even. Thus, Theorems 1 and 2 are special cases of Theorem 3.
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2. Proof of Theorem 3 To prove Theorem 3, we shall need the following lemmas due to Henning [7]. Observing that the complete graph Kn is 1-factorable for n being even, we immediately have Lemma 1 (Henning [7]). If k and n are integers with k < n and n is even, then we can construct a k-regular graph on n vertices. Lemma 2 (Henning [7]). A signed total dominating function of a graph G = (V, E) is minimal if and only if for every vertex v ∈ V with f (v) = 1, there exists a vertex u ∈ N(v) with f [u] ≤ 2, where if deg(u) is odd, then f [u] = 1; if deg(u) is even, then f [u] = 2. Now we can present the proof of Theorem 3. Proof of Theorem 3. Let f be a Γts (G)-function of G, and let P and M denote the sets of those vertices in G that are assigned under f the value +1 and −1, respectively. If δ = 1 or 2, then the results are trivial. Hence in what follows we assume that δ ≥ 3. For notational convenience, we write b(δ − 1)/2c = s1 , d(δ + 1)/2e = s2 , b(∆ − 1)/2c = t1 and d(∆ + 1)/2e = t2 . Obviously, f [v] = deg(v) − 2degM (v) for any vertex v ∈ V . Then, by Lemma 2, M 6= ∅. Let |P| = p and |M| = m. Thus, w(f ) = |P | − |M| = n − 2m. For any vertex v ∈ V , degM (v) ≤ b(deg(v) − 1)/2c for otherwise f [v] < 1. Hence we can partition P into t1 + 1 sets by defining Pi = {v ∈ P | degM (v) = i} and letting |Pi | = pi for i = 0, 1, . . . , t1 . Then we have n=m+p=m+
t1 X
pi .
(1)
i =0
For any vertex v ∈ V , degP (v) ≥ d(deg(v) + 1)/2e for otherwise f [v] < 1. We define Mj = {v ∈ M | degP (v) = j}
for j = s2 , s2 + 1, . . . , t2 Pt2 0 and M0 = M − j=s2 Mj . Let |Mj | = mj , and so |M0 | = m − j= s2 mj . Clearly, (Ms2 , . . . , Mt2 , M ) is a partition of M. Since each 0 vertex in M is adjacent to at most ∆ vertices of P, we have St2
t1 X
ipi = e(P, M) ≤ (s2 ms2 + · · · + t2 mt2 ) + ∆ m − (ms2 + · · · + mt2 )
.
i =1
Hence, t1 X
ipi ≤ ∆m −
(∆ − s2 )ms2 + · · · + (∆ − t2 )mt2 .
(2)
i =1
If P0 = ∅, then, by (1) and (2), we have n=m+
t1 X i=1
pi ≤ m +
t1 X
ipi ≤ (∆ + 1)m.
i =1
This implies that m ≥ n/(∆ + 1), and so Γts (G) = n − 2m ≤ (∆ − 1)n/(∆ + 1). Observing that ∆(δ + 3) − (δ − 1) ∆(δ + 2) − (δ − 2) (∆ − 1)n/(∆ + 1) ≤ min n, n , ∆(δ + 3) + (δ − 1) ∆(δ + 2) + (δ − 2) then the desired result follows. We therefore may assume that P0 6= ∅. Ss1 −1 0 According to our partition for P and M, we have f [v] ≥ 3 for any v ∈ ( i= 0 Pi ) ∪ M , so if f [v] = 1 or 2 for v ∈ V , then St1 St2 v ∈ ( i=s1 Pi ) ∪ ( j=s2 Mj ). For any vertex v ∈ P0 , since f is minimal, by Lemma 2, there exists a vertex u ∈ N(v) such that St1 f [u] ∈ {1, 2}. Let I = {u ∈ N(P0 ) | f [u] = 1 or 2}. Then I ⊆ i= s1 Pi . So p0 = |P0 | ≤ e(P0 , I) = e P0 ,
t1 [
!
(Pi ∩ I) ≤ e P0 ,
i=s1
t1 [
! Pi
.
(3)
i=s1
Furthermore, for every vertex u ∈ Pi ∩ I (s1 ≤ i ≤ t1 ), there must exist a neighbor u0 of u such that f [u0 ] = 1 or 2. Note that u0 St1 St2 St1 St2 0 0 belongs to ( i= s1 Pi ) ∪ ( j=s2 Mj ). If u ∈ i=s1 Pi , then u is adjacent to at most i + 1 vertices of P0 , while if u ∈ j=s2 Mj , then u is adjacent to at most i + 2 vertices of P0 . Hence we can write Pi ∩ I (s1 ≤ i ≤ t1 ) as the disjoint union of two sets IPi0 and IPi00 , where IPi0 = {u ∈ Pi ∩ I | degP0 (u) = i + 2} and IPi00 = Pi ∩ I − IPi0 . Then IPi00 = {u ∈ Pi ∩ I | degP0 (u) ≤ i + 1}. Let |IPi0 | = p0i , and St1 St2 0 so |IPi00 | = |Pi ∩ I| − p0i . Since each vertex u ∈ i= s1 IPi is adjacent to a vertex in j=s2 Mj , it follows that t1 X i=s1
p0i ≤ s2 ms2 + (s2 + 1)m(s2 +1) + · · · + t2 mt2 .
(4)
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Thus, by Eqs. (3) and (4), we have p0 ≤
t1 X
(i + 2)p0i +
t1 X
(i + 2)p0i +
t1 X
t1 X
(i + 1)(pi − p0i )
i=s1
i=s1
≤
(i + 1)(|Pi ∩ I| − p0i )
i=s1
i=s1
≤
t1 X
(i + 1)pi +
t2 X
jmj .
(5)
j=s2
i=s1
We now distinguish two possibilities depending on the parity of the minimum degree δ. Case 1. δ is odd. Then s1 = (δ − 1)/2. Noting that (δ + 3)i/(δ − 1) ≥ i + 2 holds when i ≥ (δ − 1)/2, then, by Eqs. (1), (2) and (5), we obtain ! t2 t1 t1 X X X jmj + (i + 1)pi + pi n ≤ m+ i=s1
= m+
t1 X i=s1
(i + 2)pi +
i =1
j=s2 sX 1 −1
pi +
i =1
t2 X
jmj
j=s2
≤ m+
t1 sX t2 1 −1 X δ+3 X ipi + pi + jmj δ − 1 i=s1 j=s2 i=1
≤ m+
t1 t2 X δ+3 X ipi + jmj δ − 1 i =1 j=s2
≤ m+
t2 t2 X δ+3 X δ+3 jmj m∆ − (∆ − j)mj + δ−1 δ − 1 j=s2 j=s2
= m+
t2 X (δ + 3)∆ − 2j(δ + 1) δ+3 m∆ − mj . δ−1 δ−1 j=s2
Further, note that
(δ + 3)∆ − 2j(δ + 1) (δ + 3)∆ − 2t2 (δ + 1) ≥ ≥ 0, δ−1 δ−1 the sign in the above second inequality can be justified by observing the fact that t2 = (∆ + 1)/2 if ∆ = δ and t2 ≤ (∆ + 2)/2 if ∆ ≥ δ + 1. Then n ≤ m + ((δ + 3)∆/(δ − 1))m, which implies that m ≥ (δ − 1)n/(∆(δ + 3) + (δ − 1)). Hence, ∆(δ + 3) − (δ − 1) Γts (G) = w(f ) = n − 2m ≤ n. ∆(δ + 3) + (δ − 1) That the bound is sharp may be seen as follows. For any integers l ≥ 1 and r ≥ l + 1, let Fl,r be the graph with vertex set X ∪ Y ∪ Z with |X | = l, |Y | = 2r and |Z | = 2r(l + 1), where X is an independent set of vertices. The edge set of Fl,r is constructed as follows: Add 2rl edges between X and Y so that hX ∪ Y i forms a complete bipartite graph with partition sets X and Y . Add 2r(l + 1) edges between Y and Z so that each vertex of Y is precisely adjacent to l + 1 vertices of Z and each vertex of Z is precisely adjacent to one vertex of Y . Add edges joining vertices of Y so that Y induces a 1-regular graph. Add edges joining vertices of Z so that Z induces a 2l-regular graph (since 2l < 2r(l + 1) = |Z | and |Z | is even, it follows from Lemma 1 that such an addition of edges is possible). By construction, Fl,r is a graph of order n = l + 2r + 2r(l + 1) with minimum degree δ = 2l + 1 and maximum degree ∆ = 2r. The function g that assigns to each vertex of X the value −1 and to each vertex of Y ∪ Z the value +1 is a minimal signed total dominating function of Fl,r as g[u] = 2 for any u ∈ Y . It is easy to see that ∆(δ + 3) − (δ − 1) w(g) = 2r + 2r(l + 1) − l = n. ∆(δ + 3) + (δ − 1) Consequently, Γts (Fl,r ) = (∆(δ + 3) − (δ − 1))n/(∆(δ + 3) + (δ − 1)). Now suppose that G is an odd-degree graph. We shall improve the above bound. Since all vertices in G have odd degree, it follows that for any vertex v ∈ V , f [v] = deg(v) − 2degM (v) is odd. Thus, for every vertex u ∈ Pi ∩ I (s1 ≤ i ≤ t1 ), by Lemma 2, St1 St2 0 there exist u0 ∈ N(u) such that f [u0 ] = 1. If u0 ∈ i= s1 Pi , then u is adjacent to at most i vertices of P0 , while if u ∈ j=s2 Mj , then u is adjacent to at most i + 1 vertices of P0 . Hence we can partition Pi ∩ I (s1 ≤ i ≤ t1 ) into two sets IPi0 and IPi00 , where IPi0 = {u ∈ Pi ∩ I | degP0 (u) = i + 1} and IPi00 = {u ∈ Pi ∩ I | degP0 (u) ≤ i}. Let |IPi0 | = p0i , and so |IPi00 | = |Pi ∩ I| − p0i . Therefore, by
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(3) and (4), we can obtain p0 ≤
t1 X
(i + 1)p0i +
t1 X
i(|Pi ∩ I| − p0i ) ≤
ipi +
i=s1
i=s1
i=s1
t1 X
t2 X
jmj .
(6)
j=s2
Since (δ + 1)/(δ − 1)i ≥ i + 1 when i ≥ (δ − 1)/2, it follows from (1), (2) and (6) that ! t2 t1 t1 X X X δ+1 jmj + pi ≤ 1 + ipi + n≤m+ ∆ m. δ−1 j=s2 i=s1 i =1 Consequently, Γts (G) = n − 2m ≤
∆(δ + 1) − (δ − 1)
∆(δ + 1) + (δ − 1)
n.
That the bound is sharp may be seen as follows. For any integers l ≥ 1, r ≥ l and an even integer q, where 2l ≤ q ≤ 2r, let Hl,r be the graph with vertex set X ∪ Y ∪ Z with |X | = 2l2 , |Y | = 2l(2r + 1) and |Z | = 2l2 (2r + 1), where X is an independent set of vertices. The edge set of Hl,r is constructed as follows: Add 2l2 (2r + 1) edges between X and Y so that each vertex in X has degree 2r + 1 while each vertex in Y has degree l. Add 2l2 (2r + 1) edges between Y and Z so that each vertex of Y is precisely adjacent to l vertices of Z and each vertex of Z is precisely adjacent to one vertex of Y . Add l(2r + 1) edges joining vertices of Y so that Y induces a 1-regular graph. Add edges joining vertices of Z so that Z induces a q-regular graph (since q ≤ 2r < |Z | and |Z | is even, it follows from Lemma 1 that such an addition of edges is possible). By construction, Hl,r is a graph of order n = 2l2 + 2l(2r + 1) + 2l2 (2r + 1) with minimum degree δ = 2l + 1 and maximum degree ∆ = 2r + 1. The function g that assigns to each vertex of X the value −1 and to each vertex of Y ∪ Z the value +1 is a minimal signed total dominating function of Hl,r as g[u] = 1 for any u ∈ Y . It is easy to see that ∆(δ + 1) − (δ − 1) w(g) = 2l(2r + 1) + 2l2 (2r + 1) − 2l2 = n. ∆(δ + 1) + (δ − 1) Consequently, Γts (Hl,r ) = (∆(δ + 1) − (δ − 1))n/(∆(δ + 1) + (δ − 1)). Case 2. δ is even. Then s1 = (δ − 2)/2. Noting that (δ + 2)i/(δ − 2) ≥ i + 2 holds when i ≥ (δ − 2)/2, then, by Eqs. (1), (2) and (5) again, we have ! t1 t2 t1 X X X n ≤ m+ (i + 1)pi + jmj + pi i=s1
= m+
t1 X
(i + 2)pi +
i=s1
i =1
j=s2 sX 1 −1
pi +
i =1
t2 X
jmj
j=s2
≤ m+
t2 sX t1 1 −1 X δ+2 X jmj pi + ipi + δ − 2 i=s1 j=s2 i =1
≤ m+
t1 t2 X δ+2 X ipi + jmj δ − 2 i =1 j=s2
≤ m+
t2 t2 X δ+2 δ+2 X m∆ − (∆ − j)mj + jmj δ−2 δ − 2 j=s2 j=s2
= m+
t2 X δ+2 (δ + 2)∆ − 2jδ m∆ − mj . δ−2 δ−2 j=s2
Further, observing that
(δ + 2)∆ − 2jδ (δ + 2)∆ − 2t2 δ ≥ ≥ 0, δ−2 δ−2 we have n ≤ m + ((δ + 2)∆/(δ − 2))m, which implies that m ≥ (δ − 2)n/(∆(δ + 2) + (δ − 2)). Hence, ∆(δ + 2) − (δ − 2) Γts (G) = w(f ) = n − 2m ≤ n. ∆(δ + 2) + (δ − 2) That the bound is sharp may be seen as follows. For any integers l ≥ 1, r ≥ l + 1 and q, where 2l + 1 ≤ q ≤ 2r − 1, let Gl,r be the graph with vertex set X ∪ Y ∪ Z with |X | = l, |Y | = 2r and |Z | = 2r(l + 1), where X is an independent set of vertices. The edge set of Gl,r is constructed as follows: Add 2rl edges between X and Y so that hX ∪ Y i forms a complete bipartite graph with partition sets X and Y . Add 2r(l + 1) edges between Y and Z so that each vertex of Y is precisely adjacent to l + 1 vertices
E. Shan, T.C.E. Cheng / Discrete Applied Mathematics 157 (2009) 1098–1103
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Fig. 1. The graph Gl,r .
of Z and each vertex of Z is precisely adjacent to one vertex of Y . Add edges joining vertices of Y so that Y induces a 1-regular graph. Add edges joining vertices of Z so that Z induces a q-regular graph (since q < 2r(l + 1) = |Z | and |Z | is even, it follows from Lemma 1 that such an addition of edges is possible). The graph Gl,r is shown in Fig. 1. By construction, Gl,r is a graph of order n = l + 2r + 2r(l + 1) with minimum degree δ = 2(l + 1) and maximum degree ∆ = 2r. The function g that assigns to each vertex of X the value −1 and to each vertex of Y ∪ Z the value +1 is a minimal signed total dominating function of Gl,r as g[u] = 2 for any u ∈ Y . It is easy to see that ∆(δ + 2) − (δ − 2) w(g) = 2r + 2r(l + 1) − l = n. ∆(δ + 2) + (δ − 2) Consequently, Γts (Gl,r ) = (∆(δ + 2) − (δ − 2))n/(∆(δ + 2) + (δ − 2)).
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