The decycling number of outerplanar graphs

DOI 10.1007/s10878-012-9455-1

The decycling number of outerplanar graphs

Huilan Chang· Hung-Lin Fu · Min-Yun Lien

Published online: 25 February 2012

© Springer Science+Business Media, LLC 2012

Abstract For a graph G, let τ (G) be the decycling number of G and c(G) be the

number of vertex-disjoint cycles of G. It has been proved that c(G)≤ τ(G) ≤ 2c(G) for an outerplanar graph G. An outerplanar graph G is called lower-extremal if τ (G)= c(G) and upper-extremal if τ(G) = 2c(G). In this paper, we provide a nec-essary and sufficient condition for an outerplanar graph being upper-extremal. On the other hand, we find a classS of outerplanar graphs none of which is lower-extremal and show that if G has no subdivision of S for all S∈ S, then G is lower-extremal.

Keywords Decycling number· Feedback vertex number · Cycle packing number ·

Outerplanar graph

1 Introduction

The problem of destroying all cycles in a graph by deleting a set of vertices origi-nated from applications in combinatorial circuit design (Johnson1974). Also, it has found applications in deadlock prevention in operating systems (Wang et al.1985; Silberschatz et al.2003), the constraint satisfaction problem and Bayesian inference in artificial intelligence (Bar-Yehuda et al.1998), monopolies in synchronous dis-tributed systems (Peleg1998,2002), the converters’ placement problem in optical networks (Kleinberg and Kumar1999), and VLSI chip design (Festa et al.2000).

This research is partially supported by NSC 99-2811-M-009-056 and NSC 100-2115-M-390-004-MY2.

H. Chang (



Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30010, Taiwan, ROC

e-mail:huilan0102@gmail.com

H. Chang

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC

In the literature, a set of vertices of a graph whose removal leaves an acyclic graph is referred to as a decycling set (Beineke and Vandell1997), or a feedback vertex set (Wang et al.1985), of the graph. The minimum cardinality of a decycling set of G, denoted by τ (G), is referred to as the decycling number of G. Determining the decycling number is equivalent to finding the greatest order of an induced forest of Gproposed first by Erdös et al. (1986). The problem of determining the decycling number has been proved to be NP-complete for general graphs (Karp et al.1975), which also shows that even for planar graphs, bipartite graphs and perfect graphs, the computation complexity of finding their decycling numbers is not reduced.

Besides searching for the value (or an upper bound) of the decycling number in the order of a graph, another parameter that is closely related to the decycling number is the cycle packing number, which is the maximum number of vertex-disjoint cycles. We denote this parameter by c(G). Determining the cycle packing number of a graph is also known to be NP-complete (Bodlaender1994). A trivial relation between the decycling number and the cycle packing number is c(G)≤ τ(G).

A graph is said to be outerplanar provided that all its vertices lie on the boundary of a face (after embedding the graph in a sphere). Even for an outerplanar graph G, not much is known about τ (G). Bau et al. (1998) found formulas of decycling num-bers for subclasses of outerplanar graphs. For maximal outerplanar graph of order n, they provided a sharp upper boundn₃, which can be derived by the acyclic coloring argument (Fertin et al.2002). Kloks et al. (2002) proved that τ (G)≤ 2c(G) by a greedy algorithm.

An outerplanar graph G is called lower-extremal if τ (G)= c(G) and upper-extremal if τ (G)= 2c(G). In this paper, we provide a necessary and sufficient con-dition for an outerplanar graph being upper-extremal. On the other hand, we provide a sufficient condition for an outerplanar graph being lower-extremal. We find a class S of outerplanar graphs none of which is lower-extremal and show that if G has no subdivision of S (or S-subdivision) for all S∈ S, then G is lower-extremal.

For graphs notations and terminologies here, we refer to West (2001).

2 Upper-extremal graphs

For simplicity, we use ij to denote an edge{i, j}. We start by presenting an upper-extremal graph with simplest structure.

Definition 1 Sk is a graph with vertex set V = {0, 1, . . . , 2k − 1} and edge set E =

{i(i + 1) : 0 ≤ i ≤ 2k − 1} ∪ {i(i + 2) : i is even} (the indices are under modulo 2k).

Then τ (Sk)= k2 and c(Sk)= 

k

2. S3is clearly an upper-extremal graph; indeed, its subdivisions are the only 2-edge-connected outerplanar graphs that are upper-extremal and have cycle packing number one. We define the simplified graph of a graph G to be the graph obtained from G by continuously deleting isolated vertices or degree one vertices until there is no more such vertex and denote it byG.

Throughout the paper, let F (G) denote the outer face of an outerplanar G. An edge uv is called a basic edge of G if uv and some u, v-path on the boundary of F (G)form the boundary of a face of G. Then, we have

Fig. 1 An S3-tree G of order 3,

where τ (G)= 6 = 2c(G)

Lemma 1 For an outerplanar graph G with c(G)= 1, G is upper-extremal if and

only ifG is an S3-subdivision.

Proof It suffices to prove the necessity. IfG has a cut-vertex v, then v belongs to two blocks ofG, say G1and G2, andG − v has a cycle which is vertex-disjoint with G1or G2. ThenG has two vertex-disjoint cycles, a contradiction. Thus G is 2-connected. Any two basic edges ofG have a common vertex; otherwise, we can find two vertex-disjoint cycles. This implies thatG has at most three basic edges. ThenG has exactly three basic edges; otherwise we can decycle it by deleting one

vertex. Hence it is an S3-subdivision. 

To characterize the extremal graphs, we first define a class of special upper-extremal graphs—S3-trees. A graph is an S3-tree of order t if it has exactly t vertex-disjoint S3-subdivisions and every edge not on these S3-subdivisions belongs to no cycle (see Fig.1for an example). It is easy to verify that any S3-tree of order t has exactly t vertex-disjoint cycles, and to decycle an S3-tree, we have to delete two vertices from each S3-subdivision. Hence, all S3-trees are upper-extremal. We will show that there is no other upper-extremal outerplanar graph.

For X, Y⊆ V (G), an X, Y -path is a path having one endpoint in X, the other one in Y , and no other vertex in X∪ Y , and a {v}, Y -path is simply written as a v, Y -path. Then,

Lemma 2 An outerplanar graph G comprised of a connected S3-tree H of order t and two internally disjoint v, V (H )-paths has t + 1 vertex-disjoint cycles for v /∈ V (H).

Proof Suppose that v1, v2∈ V (H) are the endpoints of these two v, V (H)-paths. Let C be the cycle comprised of these two v, V (H )-paths and the v1, v2-path in H such that C is the boundary of some face of G. Then the intersection (vertex and edge) of C and any S3-subdivision S in H is either an edge on the boundary of the outer face of S or a vertex of S; otherwise, there would be a subdivision of K2,3or K4, a contradiction. Hence, we can easily find a cycle in every S3-subdivision that is

vertex-disjoint with C. 

Theorem 3 An outerplanar graph G is upper-extremal if and only if G is an S3-tree. Proof It suffices to consider the necessity. We prove it by induction on c(G). The statement is clearly true for G if c(G)= 0. Let G be an upper-extremal graph. Then we can find a maximal induced path P with some endpoints u and v such that uv

Fig. 2 Gray edges form some vertex-disjoint cycles

is an edge of G (u= v since G is extremal). Then G \ {u, v} must be upper-extremal and c(G\ {u, v}) ≤ c(G) − 1. Thus we can assume that G \ {u, v} is an S3-tree of order t . Then c(G)≥ t + 1. Since τ(G) ≤ 2t + 2 and G is upper-extremal, c(G)= t + 1 and thus τ(G) = 2t + 2.

Define G∗:= G \ {x : x is on some cycle of G \ {u, v} }. Then c(G∗)= 1. If τ (G∗)= 2, then by Lemma1 G∗ is an S3-tree of order one. This implies that G contains t+ 1 vertex-disjoint S3-subdivisions. By Lemma2, there exists at most one path between any two S3-subdivisions and thus G is an S3-tree. Now, we consider w.l.o.g. that G∗− u is acyclic. Let V∗:= V (G∗). Then G is a graph comprised of G∗,G \ V∗, and some internally disjoint V∗, V (G \ V∗)-paths. Notice that there is at most one w, V∗-path if w∈ V (G \ V∗) is not on any S3-subdivision. We classify the vertices in V∗\ V (P ) into two disjoint sets A and B where A is the union of the vertex sets of components of G∗− u except the one containing v. Let V be the vertex set of a component ofG \ V∗. Then each component of G[A] has at most one path to V and there is at most one B, V -path; otherwise, by Lemma2

c(G)≥ t + 2 (see Fig.2(a)), a contradiction. We consider the following cases. Case 1: G∗has a cycle containing u but not v. Then there is at most one v, V -path; otherwise, c(G)≥ t + 2. Then the remaining case we have to deal with is that there is exactly one B, V -path and one u, V -path. Let x, y be the endpoints of these two paths in V . Then at least one of x and y is on an S3-subdivision in G[V ] and thus we can decycle G by deleting u and a minimum decycling set of G\ {u, v} including it, contradicting the fact that τ (G)= 2t + 2.

Case 2: Every cycle of G∗ contains both u and v. Then G∗− v is also acyclic. Suppose that Vu ⊆ V is the set of vertices as the endpoints of some u,V -paths

and Vv⊆ V is the set of vertices as the endpoints of some B∪ {v},V -paths. If

min(|Vu|, |Vv|) ≥ 2 and max(|Vu|, |Vv|) ≥ 3, then by Lemma2 c(G)≥ t + 2 (see Fig. 2(b) for an example), a contradiction. Thus |Vu| = 2 = |Vv| or |Vu| = 1 or

|Vv| = 1. If |Vu| = 1 (or |Vv| = 1), then G can be decycled by deleting v (or u) and a

minimum decycling set of G\ {u, v}, contradicting that τ(G) = 2t + 2. It remains to consider that|Vu| = 2 = |Vv|. If Vu∩ Vv= ∅, then c(G) ≥ t + 2 (see Fig.2(c) for an example), a contradiction. Suppose that Vu∩ Vv= {w}. Then w must be on some S

3-subdivision. Therefore, we can decycle G by deleting u and a minimum decycling set of G\ {u, v} with w included (see Fig.2(d) for an example), again a contradiction.

3 Lower-extremal graphs

To prove that a property is sufficient for a graph being lower-extremal, we will use induction. In order to facilitate the proof of the induction step, we need a hereditary graph property. A graph property is called monotone if it is closed under removal of vertices. We provide the following general result that is applicable to all graphs.

Lemma 4 Suppose that a 2-connected graph is lower-extremal provided that it

sat-isfies a monotone propertyP. Then G is lower-extremal if G satisfies P.

Proof We prove the statement by induction on|G|. The statement is true for graphs with c(G)= 0 or |V (G)| = 1. For a graph G of connectivity one, let G1 be a leaf block of G and v be the cut-vertex of G in V (G1). Let G2= G \ V (G1− v). Then c(G)is either c(G1)+ c(G2)or c(G1)+ c(G2)− 1, and τ(G) ≤ τ(G1)+ τ(G2). Thus suppose to the contrary that τ (G) > c(G). Then c(G)= c(G1)+ c(G2)− 1 and τ (G)= τ(G1)+ τ(G2). The first equality shows that every maximum set of vertex-disjoint cycles of Gi must contain a cycle with v for i= 1, 2, and thus c(Gi− v) <

c(Gi)for i= 1, 2. The second equality shows that v does not belong to any minimum

decycling set of G∗where G∗= G1or G2and thus τ (G∗− v) = τ(G∗). Thus by the monotonicity ofP and the induction hypothesis, c(G∗− v) = τ(G∗− v) = τ(G∗)=

c(G∗), a contradiction. 

To introduce a sufficient condition for a graph being lower-extremal, we first clas-sify all edges of an outerplanar graph. For a 2-connected outerplanar graph G, let E0(G)and E1(G)be the set of edges on the boundary of F (G) and the set of ba-sic edges of G, respectively. For k≥ 2, define Ek(G)to be the set of basic edges of G\k_i=1−1Ei(G). For an edge uv∈ Ek(G), we use C(uv) to denote a cycle generated

by uv and a u, v-path on the boundary of F (G) such that the cycle is the boundary of a face of G\k_i=1−1Ei(G). We also call it a basic cycle of the graph G\

k−1 i=1Ei(G)

generated from edge uv.

Lemma 5 If G is a 2-connected outerplanar graph with no Sk-subdivision for all odd number k, then G is lower-extremal.

Proof We prove the statement by induction on|E(G)|. It is easy to verify that the statement is true for graphs with at most three edges. It suffices to prove that there exists a 2-connected subgraph G of G that has fewer number of edges and no Sk

-subdivision for all odd number k and satisfies τ (G)≤ τ(G )(then τ (G)≤ τ(G )= c(G )≤ c(G)).

The statement is clearly true for G with |E2(G)| = 0. Suppose |E2(G)| ≥ 1 (and thus|E1(G)| ≥ 1). Take an edge e = xy ∈ E2(G) and a basic cycle C(e) of G\ E1(G). Let E⊆ E1(G) be the set of edges with both endpoints on C(e). We consider the following cases.

Case 1: E induces an x, y-path of G, say xv1v2· · · vty. Here, t must be even since G contains an St+2-subdivision. Let D be a minimum decycling set of

G− e. If D contains x or y, then D is also a decycling set of G and thus τ (G)≤ τ(G − e). Suppose x, y /∈ S. W.l.o.g., we can assume that D ∩ C(e) con-tains only vertices of degree larger than two. Then |D ∩ C(e)| ≥ (t + 2)/2. Let D = (D \ C(e)) ∪ {x, v2, v4, . . . , vt}. Then D is a decycling set of G of size at

most τ (G− e). Thus, τ(G) ≤ τ(G − e).

Case 2: E generates a maximal path that contains none of x and y, say v1v2· · · vt. We let G to denote G\ V (C(e) − x − y) if E = {vivi+1: i = 1, . . . , t − 1} and

G\ {vivi+1: i = 1, . . . , t − 1} otherwise. Then G is clearly 2-connected. Thus we

have τ (G)≤ τ(G )+ ₂t = c(G )+ ₂t = c(G).

Case 3: E induces at most two components which are paths as xv1v2· · · vt and yu1u2· · · ut . Suppose t (or t ) is odd. Let D be a minimum decycling set of G− e. Similar to the argument in Case 1, suppose that x, y /∈ D. Then |D ∩

{v1, v2, . . . , vt}| ≥ (t + 1)/2 and thus (D \ {v1, v2, . . . , vt}) ∪ {x, v2, v4, . . . , vt−1} is

a decycling set of G. Hence τ (G)≤ τ(G − e). It remains to consider that t and t are even. Let G = G \ V (C(e) − x − y) and D be a minimum decycling set of G . Then D∪ {v1, v3, . . . , vt−1} ∪ {u1, u2, . . . , ut −1} is a decycling set of G of size

τ (G )+ (t + t )/2. Since G[V (C(e))] has (t + t )/2 vertex-disjoint cycles that do not contain x and y, τ (G)≤ τ(G )+ (t + t )/2= c(G )+ (t + t )/2≤ c(G). This

concludes the proof. 

The property of being without Sk-subdivision is monotone. Therefore, by

Lemma4and Lemma5, we have

Theorem 6 For an outerplanar graph G, if G has no Sk-subdivision for all odd number k, then G is lower-extremal.

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