- Email: [email protected]
On spanning trees with restricted degrees
Information Processing Letters 73 (2000) 163–165
On spanning trees with restricted degrees Atsushi Kaneko a,1 , Kiyoshi Yoshimoto b,∗ a Department of Computer Science and Communication Engineering, Kogakuin University, 1-24-2 Nishi-Shinjuku, Shinjuku-ku,
Tokyo, 163-8677 Japan b Department of Mathematics, College of Science and Technology, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku,
Tokyo, 101-8308 Japan Received 25 October 1999; received in revised form 6 January 2000 Communicated by K. Iwama
Abstract Let G be a connected graph and X be a vertex subset of G. Let f be a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. In this paper, we show some conditions for graphs to contain a spanning tree such that f (x) 6 degT (x) for all x ∈ X. 2000 Elsevier Science B.V. All rights reserved. Keywords: Combinatorial problems; Spanning trees
In this paper, we denote by NG (x) the set of vertices which is adjacent to x in a graph G and its cardinality by degG (x). The subgraph of G induced by a vertex subset S is denoted by hSi and the union of NG (x) for all x ∈ S is simply denoted by NG (S). By a matching M from A to B, we mean one such that every edge of M joins a vertex of A to a vertex of B and every vertex of A is incident with an edge of M. A bipartite complete graph is said to be balanced if its partite sets have the same order. All notation and terminology not explained here are given in [3]. There are various results about factors of graphs [1]. We discuss spanning trees of graphs which satisfy given degree conditions. Let X be a vertex subset of a connected graph G and f a mapping from X to the set of natural numbers such that f (x) > 2. What kind of conditions will be needed to ensure that a graph G ∗ Corresponding author. Email: [email protected].
has a spanning tree T such that f (x) 6 degT (x) for all x ∈ X? At first we shall show the following theorem. Theorem 1. Let G be a connected bipartite graph with partite sets X and Y and f a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. There exists a spanning tree T such that f (x) 6 degT (x) for all x ∈ X if and only if it holds that, for any subset S ⊂ X, X NG (S) > f (x) − |S| + 1. x∈S
Proof. If there is a spanning tree T such that f (x) 6 degT (x) for all x ∈ X, then, for any S ⊂ X, we have X
f (x) − |S| + 1 6
x∈S
jp. 1 Email: [email protected].
0020-0190/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 0 0 ) 0 0 0 1 8 - 1
X x∈S
degT (x) − |S| + 1
6 NT (S) 6 NG (S) .
164
A. Kaneko, K. Yoshimoto / Information Processing Letters 73 (2000) 163–165
Fig. 1.
Thus let us show the if part. Assume that X and f satisfy the sufficient condition. We first construct a graph as follows. For all x ∈ X, we recursively remove x and add new f (x) − 1 vertices Z(x) and edges {yz | y ∈ NG (x), z ∈ Z(x)}. The S resultant graph H is bipartite with the partite sets x∈X Z(x) and Y . Let S ϕ be a mapping from x∈X Z(x) to X such that an element of Z(x) correspond to x. Then we have X f (x) − 1 |S| 6 x∈ϕ(S)
<
X
f (x) − ϕ(S) + 1
The desired spanning tree can be computed in polynomial time by using an efficient matching algorithm. Next we consider a spanning tree of a general graph G, i.e., let X be a vertex subset of G and f a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. Let {Ci } be the connected components in the induced subgraph hXi. A connected component Ci contains at most |V (Ci )| − 1 edges in any spanning tree of G. Thus |NG (Ci ) \ X| must be greater than or equal to X f (x) − 2 |V (Ci )| − 1 . x∈Ci
We denote by ω(S) the number of the connected components in hSi. Then it holds that X X f (x) − 2 |V (Ci )| − 1 i6ω(X)
=
X
x∈Ci
f (x) − 2|X| + 2ω(X).
x∈X
x∈ϕ(S)
6 NG (ϕ(S)) = NH (S) S for any SS ⊂ x∈X Z(x). Thus there exists a matching from x∈X Z(x) to Y by Hall’s theorem [4]. Let F be the forest obtained from the matching by identifying each vertex set Z(x) with corresponding vertex x for all x ∈ X, so that degF (x) = f (x)−1. See Fig. 1. Any connected component of F is a star R(x) with some center x ∈ X. Then, there exists a forest F such that degF (x) = f (x) for all x ∈ X as follows. We show it by an induction on the number of vertices with degree f (x) − 1 in a forest. Let F 0 be a forest which contains F such that f (x) − 1 6 degF 0 (x) 6 f (x) for all x ∈ X and let X0 be the set of all the vertices of which the degree in F 0 equals f (x) − 1. Because we have X f (x) − 1 < NG (X0 ) , x∈X 0
S there is an edge h joining X0 and Y \ V ( x∈X0 R(x)). In the forest F 0 ∪ h, the number of vertices with degree f (x) − 1 is fewer than |X0 |. Now we can obtain a desired spanning tree when we add some edges to F between connected components in F . 2
Since we can admit that NG (Ci ) \ X and NG (Cj ) \ X contain one common vertex, the authors conjecture as follows. Conjecture 1. Let X be any vertex subset in a connected graph G and f a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. If it holds that X NG (S) \ X > f (x) − 2|S| + ω(S) + 1, x∈S
for any subset S ⊂ X, then there exists a spanning tree T such that f (x) 6 degT (x) for all x ∈ X. If there is no edge in the induced subgraph of X, then we can show that the conjecture is true with Theorem 1 in the same way. On the other hand, we shall show the conjecture for f (x) ≡ 2 for all P x ∈ X as follows. Since x∈S f (x) = 2|S|, we have |NG (S) \ X| > ω(S) + 1 for any S ⊂ X. We classify a connected component Ci in the induced subgraph hXi whether an independency tree is contained. An independency tree is a spanning tree of which the end vertices are pairwise nonadjacent in G. Böhme
A. Kaneko, K. Yoshimoto / Information Processing Letters 73 (2000) 163–165
165
Fig. 2.
et al. [2] determined types of graphs which does not contain an independency tree as follows. Theorem 2 (Böhme et al.). A connected graph does not have an independency tree if and only if the graph is isomorphic to a cycle, a complete graph or a balanced bipartite complete graph. We label the index of a connected component Ci of hXi so that if Ci has an independency tree, then i 6 l. Let Ti be an independency tree of Ci for i 6 l. Since a Hamiltonian path is contained in a graph if it has no independency tree, we choose a Hamiltonian path of Ci as Ti for all i > l. Let Ai be the set of end vertices of Ti for all i and Ai = S {ui , vi } for i > l. There exists a matching M from A = i6l Ai ∪ {ui | i > l} to G − X because A is independent in G and we have |NG (S) \ X| > |S| + 1 for anyS subset S ⊂ A. See Fig. 2. If l = ω(X), then let F = i6ω(X) Ti ∪ M. Otherwise, we can choose edge S ei between vi and G − X for all i > l such that F = i6ω(X) Ti ∪ M ∪ {ei | i > l} is acyclic as follows. It is shown by an induction on the number of vertices with degree one ofS B = {vi | l < i 6 ω(X)} in a forest F 0 which contains i6ω(X) Ti ∪ M. Let wi be the vertex which is adjacent to ui in M and let B 0 be the intersection of B and the set of end vertices in F 0 . Since |NG (B 0 ) \ X| > |B 0 | + 1, there exists an edge h between B 0 and G − X − {wi | vi ∈ B 0 }. In the forest
F 0 ∪ h, the number of vertices with degree one of B is less than |B 0 |. Thus the following theorem is shown because we can add some edges to F ∪ V (G − F ) such that the resultant graph is a spanning tree of G. Theorem 3. Let G be a connected graph and X a vertex subset of G. Then, there exists a spanning tree T such that degT (x) > 2 for all x ∈ X if it holds that NG (S) \ X > ω(S) + 1 for any subset S ⊂ X. By the carefully examination of the proof of Theorem 2 in [2], an independency tree can be found in polynomial time. Therefore, we also obtain the desired spanning tree in Theorem 3 in polynomial time. References [1] J. Akiyama, M. Kano, Factors and factorizations of graphs— A survey, J. Graph Theory 9 (1985) 1–42. [2] T. Böhme, H.J. Broersma, F. Göbel, A.V. Kostochka, M. Stiebitz, Spanning trees with pairwise nonadjacent endvertices, Discrete Math. 170 (1997) 219–222. [3] G. Chartrand, L. Lesniak, Graphs & Digraphs, 2nd edn., Wadsworth & Brooks/Cole. [4] P. Hall, On representation of subsets, J. London Math. Soc. 10 (1935) 26–30.
On spanning trees with restricted degrees Atsushi Kaneko a,1 , Kiyoshi Yoshimoto b,∗ a Department of Computer Science and Communication Engineering, Kogakuin University, 1-24-2 Nishi-Shinjuku, Shinjuku-ku,
Tokyo, 163-8677 Japan b Department of Mathematics, College of Science and Technology, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku,
Tokyo, 101-8308 Japan Received 25 October 1999; received in revised form 6 January 2000 Communicated by K. Iwama
Abstract Let G be a connected graph and X be a vertex subset of G. Let f be a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. In this paper, we show some conditions for graphs to contain a spanning tree such that f (x) 6 degT (x) for all x ∈ X. 2000 Elsevier Science B.V. All rights reserved. Keywords: Combinatorial problems; Spanning trees
In this paper, we denote by NG (x) the set of vertices which is adjacent to x in a graph G and its cardinality by degG (x). The subgraph of G induced by a vertex subset S is denoted by hSi and the union of NG (x) for all x ∈ S is simply denoted by NG (S). By a matching M from A to B, we mean one such that every edge of M joins a vertex of A to a vertex of B and every vertex of A is incident with an edge of M. A bipartite complete graph is said to be balanced if its partite sets have the same order. All notation and terminology not explained here are given in [3]. There are various results about factors of graphs [1]. We discuss spanning trees of graphs which satisfy given degree conditions. Let X be a vertex subset of a connected graph G and f a mapping from X to the set of natural numbers such that f (x) > 2. What kind of conditions will be needed to ensure that a graph G ∗ Corresponding author. Email: [email protected].
has a spanning tree T such that f (x) 6 degT (x) for all x ∈ X? At first we shall show the following theorem. Theorem 1. Let G be a connected bipartite graph with partite sets X and Y and f a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. There exists a spanning tree T such that f (x) 6 degT (x) for all x ∈ X if and only if it holds that, for any subset S ⊂ X, X NG (S) > f (x) − |S| + 1. x∈S
Proof. If there is a spanning tree T such that f (x) 6 degT (x) for all x ∈ X, then, for any S ⊂ X, we have X
f (x) − |S| + 1 6
x∈S
jp. 1 Email: [email protected].
0020-0190/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 0 0 ) 0 0 0 1 8 - 1
X x∈S
degT (x) − |S| + 1
6 NT (S) 6 NG (S) .
164
A. Kaneko, K. Yoshimoto / Information Processing Letters 73 (2000) 163–165
Fig. 1.
Thus let us show the if part. Assume that X and f satisfy the sufficient condition. We first construct a graph as follows. For all x ∈ X, we recursively remove x and add new f (x) − 1 vertices Z(x) and edges {yz | y ∈ NG (x), z ∈ Z(x)}. The S resultant graph H is bipartite with the partite sets x∈X Z(x) and Y . Let S ϕ be a mapping from x∈X Z(x) to X such that an element of Z(x) correspond to x. Then we have X f (x) − 1 |S| 6 x∈ϕ(S)
<
X
f (x) − ϕ(S) + 1
The desired spanning tree can be computed in polynomial time by using an efficient matching algorithm. Next we consider a spanning tree of a general graph G, i.e., let X be a vertex subset of G and f a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. Let {Ci } be the connected components in the induced subgraph hXi. A connected component Ci contains at most |V (Ci )| − 1 edges in any spanning tree of G. Thus |NG (Ci ) \ X| must be greater than or equal to X f (x) − 2 |V (Ci )| − 1 . x∈Ci
We denote by ω(S) the number of the connected components in hSi. Then it holds that X X f (x) − 2 |V (Ci )| − 1 i6ω(X)
=
X
x∈Ci
f (x) − 2|X| + 2ω(X).
x∈X
x∈ϕ(S)
6 NG (ϕ(S)) = NH (S) S for any SS ⊂ x∈X Z(x). Thus there exists a matching from x∈X Z(x) to Y by Hall’s theorem [4]. Let F be the forest obtained from the matching by identifying each vertex set Z(x) with corresponding vertex x for all x ∈ X, so that degF (x) = f (x)−1. See Fig. 1. Any connected component of F is a star R(x) with some center x ∈ X. Then, there exists a forest F such that degF (x) = f (x) for all x ∈ X as follows. We show it by an induction on the number of vertices with degree f (x) − 1 in a forest. Let F 0 be a forest which contains F such that f (x) − 1 6 degF 0 (x) 6 f (x) for all x ∈ X and let X0 be the set of all the vertices of which the degree in F 0 equals f (x) − 1. Because we have X f (x) − 1 < NG (X0 ) , x∈X 0
S there is an edge h joining X0 and Y \ V ( x∈X0 R(x)). In the forest F 0 ∪ h, the number of vertices with degree f (x) − 1 is fewer than |X0 |. Now we can obtain a desired spanning tree when we add some edges to F between connected components in F . 2
Since we can admit that NG (Ci ) \ X and NG (Cj ) \ X contain one common vertex, the authors conjecture as follows. Conjecture 1. Let X be any vertex subset in a connected graph G and f a mapping from X to the set of natural numbers such that f (x) > 2 for all x ∈ X. If it holds that X NG (S) \ X > f (x) − 2|S| + ω(S) + 1, x∈S
for any subset S ⊂ X, then there exists a spanning tree T such that f (x) 6 degT (x) for all x ∈ X. If there is no edge in the induced subgraph of X, then we can show that the conjecture is true with Theorem 1 in the same way. On the other hand, we shall show the conjecture for f (x) ≡ 2 for all P x ∈ X as follows. Since x∈S f (x) = 2|S|, we have |NG (S) \ X| > ω(S) + 1 for any S ⊂ X. We classify a connected component Ci in the induced subgraph hXi whether an independency tree is contained. An independency tree is a spanning tree of which the end vertices are pairwise nonadjacent in G. Böhme
A. Kaneko, K. Yoshimoto / Information Processing Letters 73 (2000) 163–165
165
Fig. 2.
et al. [2] determined types of graphs which does not contain an independency tree as follows. Theorem 2 (Böhme et al.). A connected graph does not have an independency tree if and only if the graph is isomorphic to a cycle, a complete graph or a balanced bipartite complete graph. We label the index of a connected component Ci of hXi so that if Ci has an independency tree, then i 6 l. Let Ti be an independency tree of Ci for i 6 l. Since a Hamiltonian path is contained in a graph if it has no independency tree, we choose a Hamiltonian path of Ci as Ti for all i > l. Let Ai be the set of end vertices of Ti for all i and Ai = S {ui , vi } for i > l. There exists a matching M from A = i6l Ai ∪ {ui | i > l} to G − X because A is independent in G and we have |NG (S) \ X| > |S| + 1 for anyS subset S ⊂ A. See Fig. 2. If l = ω(X), then let F = i6ω(X) Ti ∪ M. Otherwise, we can choose edge S ei between vi and G − X for all i > l such that F = i6ω(X) Ti ∪ M ∪ {ei | i > l} is acyclic as follows. It is shown by an induction on the number of vertices with degree one ofS B = {vi | l < i 6 ω(X)} in a forest F 0 which contains i6ω(X) Ti ∪ M. Let wi be the vertex which is adjacent to ui in M and let B 0 be the intersection of B and the set of end vertices in F 0 . Since |NG (B 0 ) \ X| > |B 0 | + 1, there exists an edge h between B 0 and G − X − {wi | vi ∈ B 0 }. In the forest
F 0 ∪ h, the number of vertices with degree one of B is less than |B 0 |. Thus the following theorem is shown because we can add some edges to F ∪ V (G − F ) such that the resultant graph is a spanning tree of G. Theorem 3. Let G be a connected graph and X a vertex subset of G. Then, there exists a spanning tree T such that degT (x) > 2 for all x ∈ X if it holds that NG (S) \ X > ω(S) + 1 for any subset S ⊂ X. By the carefully examination of the proof of Theorem 2 in [2], an independency tree can be found in polynomial time. Therefore, we also obtain the desired spanning tree in Theorem 3 in polynomial time. References [1] J. Akiyama, M. Kano, Factors and factorizations of graphs— A survey, J. Graph Theory 9 (1985) 1–42. [2] T. Böhme, H.J. Broersma, F. Göbel, A.V. Kostochka, M. Stiebitz, Spanning trees with pairwise nonadjacent endvertices, Discrete Math. 170 (1997) 219–222. [3] G. Chartrand, L. Lesniak, Graphs & Digraphs, 2nd edn., Wadsworth & Brooks/Cole. [4] P. Hall, On representation of subsets, J. London Math. Soc. 10 (1935) 26–30.