Decycling Number of Linear Graphs of Trees
Jian Wang, Xirong Xu

TL;DR
This paper investigates the decycling number of line graphs of trees, providing bounds and exact values for specific classes, thereby advancing understanding of acyclic subgraph removal in these graph structures.
Contribution
It establishes bounds on the decycling number of line graphs of trees and determines exact values for perfect k-ary trees, introducing extremal constructions.
Findings
Bounds on decycling number for line graphs of trees
Exact decycling number for line graphs of perfect k-ary trees
Extremal trees achieving bounds
Abstract
The decycling number of a graph is the minimum number of vertices whose removal from results in an acyclic subgraph. It is known that determining the decycling number of a graph is equivalent to finding the maximum induced forests of . The line graphs of trees are the claw-free block graphs. These graphs have been used by Erd\H{o}s, Saks and S\'{o}s to construct graphs with a given number of edges and vertices whose maximum induced tree is very small. In this paper, we give bounds on the decycling number of line graphs of trees and construct extremal trees to show that these bounds are the best possible. We also give bounds on the decycling number of line graph of -ary trees and determine the exact the decycling number of line graphs of perfect -ary trees.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Graph theory and applications · Graph Labeling and Dimension Problems
Decycling Number of Linear Graphs of Trees
Jian Wanga, Xirong Xu*b,*111 Corresponding author: [email protected]
(*a**Department of Mathematics
Taiyuan University of Technology, Taiyuan, 030024, P.R.China
bSchool of Computer Science and Technology
Dalian University of Technology, Dalian, 116024, P.R.China *)
Abstract. The decycling number of a graph is the minimum number of vertices whose removal from results in an acyclic subgraph. It is known that determining the decycling number of a graph is equivalent to finding the maximum induced forests of . The line graphs of trees are the claw-free block graphs. These graphs have been used by Erdős, Saks and Sós to construct graphs with a given number of edges and vertices whose maximum induced tree is very small. In this paper, we give bounds on the decycling number of line graphs of trees and construct extremal trees to show that these bounds are the best possible. We also give bounds on the decycling number of line graph of -ary trees and determine the exact the decycling number of line graphs of perfect -ary trees.
Keywords: maximum induced forests, maximum linear forests, line graphs of trees, decycling number.
1 Introduction
Let be a simple graph, with vertex set and edge set . A subset is called a decycling set if the subgraph is acyclic. The minimum cardinality of a decyling set is called the decycling number (or feedback number) of proposed first by Beineke and Vandell [2]. We use the notation to denote the decycling number of .
In fact, the problem of determining the decycling number of a graph is -complete by Karp [10] (also see [6]). The best known approximation algorithm for this problem has approximation ratio [1]. Determining the decycling number is difficult even for some elementary graphs. We refer the reader to an original research paper [2] for some results. Bounds on the decycling numbers have been established for some well-known graphs, such as hypercubes [4], star graphs [12], generalized petersen graphs [7], distance graphs and circulant graphs [11].
For a graph , let be the maximum number of vertices in an induced subgraph of that is a forest. An induced forest with maximum number of vertices is called a maximum induced forest of . Determining the decycling number of a graph is equivalent to finding the maximum induced forest of , since the sum of the two numbers equals the order of .
One can also study induced trees rather than forests in graphs. Let be the size of maximum induced trees in . The problem of bounding in the connected graph was first studied by Erdős, Saks and Sós[3] thirty years ago. In their paper, Erdős, Saks and Sós studied the relationship between and several natural parameters of the graph . They were able to obtain asymptotically tight bounds on when either the number of edges or the independent number of were known. Their result showed that can be very small over graphs with vertices and edges. Given a graph , its line graph is a graph such that each vertex of represents an edge of and two vertices of are adjacent if and only if their corresponding edges share a common endpoint in . Erdős, Saks and Sós use line graphs of trees to construct graphs for which is surprisingly small. Besides, Erdős, Saks and Sós also considered the problem of estimating the size of maximum induced tree in -free graphs. They use line graphs of regular trees to construct -free graphs for which is small. Recently, Jacob Fox, Po-Shen Loh and Benny Sudakov improved the results on lower bounds of maximum induced trees in -free graphs[5].
A linear forest in a graph is a vertex disjoint union of simple paths of . A maximum linear forest in is a linear forest in with maximum number of edges. The number of edges in maximum linear forests of graph is denoted by . Define the hamiltonian completion number of graph , denoted by , to be the minimum number of edges that need to be added to make hamiltonian. The hamiltonian completion problem was introduced in 1970s by Goodman and Hedetniemi[8, 9]. Goodman and Hedetniemi[8] prove the following relation between and . For any graph with vertices, if , then ; if , then .
In this paper, we study the decycling number of line graphs of trees. We show that finding maximum induced forests in line graphs is equivalent to finding maximum linear forests in original graphs. Let be a tree on vertices with diameter . We give lower and upper bounds on as follows. If is even , then
[TABLE]
If is odd, then
[TABLE]
The extremal line graphs that achieve these bounds are also constructed.
A -ary tree is a rooted tree where within each level every node has either 0 or children. A perfect -ary tree is a -ary tree in which all leaf nodes are at the same depth. In this paper, we give bounds on decycling number of line graphs of -ary trees as follows. Let be a -ary tree on vertices. Then
[TABLE]
Moreover, we prove that if is a perfect -ary tree on vertices with height , then
[TABLE]
The rest of this paper is organized as follows. In Section 2, we show that finding maximum induced forests in line graphs is equivalent to finding maximum linear forests in original graphs. In Section 3, we give lower and upper bounds on the decycling numbers of line graphs of trees with given diameter. In Section 4, we give lower and upper bounds on the decycling number of line graphs of -ary trees.
2 Maximum Induced Forests in Line Graphs
In this section, we prove that the maximum induced forests in line graphs correspond to maximum linear forests in original graphs. Denoted by the length of the longest paths in .
Lemma 2.1**.**
A vertex-disjoint path in is longest if and only if is a maximum induced tree in line graph . A linear forest in is maximum if and only if is a maximum induced forest in line graph . Thus, and .
*Proof. *It is known that if line graphs are claw-free, then they contain no induced . So do their induced trees and induced forests. It follows that every induced tree of a line graph is an induced path and every induced forest of a line graph is an induced linear forest.
Moreover, we shall show that the line graph of a vertex-disjoint path in is an induced path in and the induced path in is also a line graph of a vertex-disjoint path in . If is a vertex-disjoint path in , in which ’s are vertices and ’s are edges of . Then, we shall show is an induced path in . Otherwise, assume that forms an edge in and . Then and share a common ending point in . We have , which contradicts with path is vertex-disjoint. Conversely, if is an induced path in line graph . Let are consecutive vertices in . Clearly, is a path in and .
Thus, a vertex-disjoint path in is longest if and only if is a maximum induced tree in and a linear forest in is maximum if and only if is a maximum induced forest in .
Clearly, linear forests of have at most edges. It implies that . Therefore, we have the following corollary.
Corollary 2.1**.**
For any graph with vertices and edges, .
3 The Decycling Number of Line Graphs of Trees
Let be a tree with vertices. An inner vertex is a vertex of degree at least two. Similarly, an outer vertex (or a leaf) is a vertex of degree one. Then the vertices of can be partitioned into the set of leaves and the set of inner vertices . The cardinality of is denoted by . For any inner vertex , let be zero if has at most two neighbors of degree less than three; let be if has neighbors of degree less than three. Then we have the following lemma.
Lemma 3.1**.**
For any tree with vertices,
[TABLE]
*Proof. *Since is a tree. By adding edges on , we get a hamiltonian graph . Since has a hamiltonian cycle , each leaf of is incident to a new edge in . For any inner vertex , if is greater than zero, then at most two edges that incident to are in . It means that at least neighbors of have degree less than or equal to two and do not adjacent to in . Each of these neighbors has to be incidence to a new edge in . Thus, at least vertices are incidence to new edges. Therefore, we have
[TABLE]
On the other hand, we can get a hamiltonian cycle by adding edges to according to the following procedure. Firstly, we choose two leaves of and add an edge between them. Let be the graph . Then contains an unique cycle formed by new edge and the unique path in . Let be the graph obtained from by contracting cycle , or . Denoted by the contracted vertex. It is easy to see that is a tree with . Now choose a leaf outside in . Let be the unique path between and . Let be the vertex in the cycle that has the smallest distance to . Let be a neighbor of in . Then by adding edges , we get a larger cycle . Now let be the graph obtained from by contracting . Then is a tree with leaves. Now choose a leaf outside from . Let be the unique path between and . Let be the vertex in the cycle that has the smallest distance to . Let be a neighbor of in . Then by adding edge , we get a larger cycle in . Do this procedure repeatedly, through each step we can get a tree from tree with leaves less than 1(see Fig.1), the procedure has to be stopped when the contracted tree has only one vertex. Then we get a hamiltonian cycle by adding edges in . Thus, .
Since any tree on vertices have edges. Then . By Lemma 2.1, we know that . Moreover, it is true that . Therefore, we have the following corollary.
Corollary 3.1**.**
For any tree on vertices,
[TABLE]
Now we introduce an operation on leaves of trees that does not decrease . For any two leaves of , suppose their neighbors are . We define Leaf-Exchange operation on as removing edge from and adding edge , the obtained tree is denoted by .
Lemma 3.2**.**
For any two leaves of , .
*Proof. *Suppose is a maximum linear forest in . Then is a linear forest in . Thus, we have .
Let be a tree on vertices. The center of a tree is the set of vertices, from which the greatest distance equals to its radius. Let be one of the center of tree . Then can be viewed as a rooted tree with root . Moreover, we can partition into sets ,, where and is the radius of . In case of no confusion, is often abbreviated as . The vertex in is called the vertex at depth . Let . Let be the diameter of and be the radius of . Let be the number of degree-two vertices in . Then, we define three family of rooted trees on vertices with diameter at most as follows.
[TABLE]
By the following three lemmas, we shall show that finding the upper bounds for on all trees is equivalence to finding that on .
Lemma 3.3**.**
For any tree on vertices with diameter , there exists a tree in such that .
*Proof. *Any tree can be viewed as a rooted tree with its center as the root. Suppose to the contrary, there exist trees that we cannot find trees with larger maximum linear forest in . Let be a counterexample with maximum. Clearly, is not in . Then, has a vertex in such that or has a vertex in such that . We split the proof into two cases as follows.
Case 1. has a vertex in such that . Assume that and . Then has one neighbor and neighbors in . Let be these neighbors in and be subtrees of with root . Let be a maximum linear forest of . Then at most two edges of are in . Since , there exists one of that is not in . Without loss of generality, we assume is not in . Then by removing edge from and adding edge , we get a new tree with . Clearly, we have since is also a linear forest of . Moreover, has more vertices than . Since is the counterexample with maximum, we know that is no longer a counterexample. Therefore, there exists a tree in such that . Then , which contradicts with that is a counterexample.
Case 2. has a vertex in such that . If , then we can get a contradiction by the same argument as in Case 1. Thus, we only need to consider the case . Then has one neighbor in and has two neighbors and in . be subtrees of with root . Let be a maximum linear forest of . Then at most two edges of are in . If is not in , by removing edge from and adding edge , we get a new tree with . We have since is also a linear forest in . Since is increased by one, is no longer a counterexample. Therefore, there exists a tree in such that . We get a contradiction. If one of and is not in , without loss of generality, we assume is not in . Then by removing edge from and adding edge , we get a new tree , which also leads to a contradiction.
Therefore, the claim holds.
Lemma 3.4**.**
For any tree in , there exists a tree in such that .
*Proof. *Suppose to the contrary, there exist counterexamples. Let be the one in with minimum. Then has at least four degree-two neighbors. Assume they are . Then by definition of , it is easy to see that subtrees with roots are all paths. Let be these paths. Clearly, is one endpoint of . Let be a maximum linear forest in . Then at least two of edges are not in . Without loss of generality, we suppose are not in . Then all edges in are in .
If one of paths has length at least two. Without loss of generality, we suppose has length at least 2. Clearly, is one endpoint of . Let be the other endpoint of and be the parent of in the rooted tree . Then by removing edge and adding edge , we get a new tree . We have since is a linear forest of . Since , there exists a tree in such that . It leads to a contradiction. If all have length one. Assume that is an edge and is an edge . Then by removing edge and adding edge , we get a new tree with , which also leads to a contradiction. Thus, the claim holds.
Lemma 3.5**.**
For any tree in with and , there exists a tree in such that .
*Proof. *Suppose is a counterexample in with maximum. Clearly, . Let be a maximum linear forest in .
Firstly, we claim has no leaves in . Otherwise, assume is a leaf in . If there is a degree-3 vertex in , say . Let be a leaf of the subtree with root . Then by Leaf-Exchange operation on , we get a new tree with and . Since is a counterexample with maximum. Then there exists a tree in such that . It follows that , a contradiction. If there is no degree-3 vertex in . Then all vertices in have degree at most two. Since and , there are at least two leaves in , say and . Then by Leaf-Exchange operation on , we get a new tree with . Since , the Leaf-Exchange operation cannot decrease the diameter. Then there exists a tree in such that . It follows that , a contradiction. Thus, has no leaves in .
Now, we know that there is at most one degree-2 vertex and no leaf in . Since , the number of degree-3 vertices in has to be greater than one. Let be degree-3 vertices in . If at least one of edges is in . Without loss of generality, we assume that is in . Let be two neighbors of in . Then at least one of edges and is not in . Suppose is not in in . Then by removing edge and adding edge , we get a new tree with . Clearly, and is in . Then there exists a tree in such that . It follows that , a contradiction. If none of is in . Then at most one edge incident to is in , and edges in each subtree with root are all in . Suppose has two neighbors in . Then by removing edge and adding edge , we get a new tree . Then is a linear forest in . It follows that and . Then there exists a tree in such that . It follows that , a contradiction.
Combining all the cases, we complete the proof.
Theorem 3.1**.**
For any tree on vertices with diameter , we have
[TABLE]
*Proof. *For the lower bounds, clearly we have . Moreover, the extremal trees that achieve these lower bounds are shown in Fig.2.
Let be the center of tree . Then can be viewed as a rooted tree with root and radius . Clearly, . Then by Lemma 3.3, 3.4 and 3.5, we know that there exists a tree such that . Therefore, we only need to consider the upper bounds on for . Now we split the proof into two cases by the parity of .
Case 1. . Let be a tree in . It is easy to see that has radius at most . Since . Then, at least two vertices in have degree two, say and . Then the subtree with root and the subtree with root are two paths. We call two leaves in these two subtrees critical leaves of and call all the other leaves non-critical leaves. Let be the tree in satisfying the following two properties as shown in Figure 3:
(1) Two critical leaves of are all at depth ;
(2) All but at most one of its leaves are at depth . If the only leaf with depth less than lie in a subtree, whose root is a degree-2 vertices in , then . If the only leaf with depth less than lie in a subtree, whose root is a degree-3 vertices in , then .
We claim for any tree in , . Let be degree-2 vertices in and be degree-3 vertices in . If , let be the third degree-2 vertex in . We arrange the subtrees of with roots from left to right in the plane. Then do Leaf-Exchange operation from a rightmost leaf to a leftmost leaf with depth less than convectively. Finally, we shall arrive the tree . Since Leaf-Exchange operation from leaf to leaf can never decrease the value of . It follows that .
Let be the remainder of dividing by . We splits the proof into two parts by the value of .
**Case 1.1. ** If , then as shown in Fig.3 (1). It is easy to see that there are three vertices in . It follows that . Moreover, the number of leaves in can be computed as follows.
[TABLE]
By Lemma 3.1, we have
[TABLE]
However, by removing the dashed edges as shown in Fig.3 (1), we get a linear forest with edges. Thus, .
**Case 1.2. ** If is [math] or between and , then is as shown in Fig.3 (2). For the second case, we have and . By Lemma 3.1, we have
[TABLE]
However, by removing the dashed edge as shown in Fig.3 (2), we get a linear forest with edges. Thus, .
Combining the two subcases, we prove that for any tree with vertices and diameter ,
[TABLE]
Case 2. . Let be a tree in . Since and . Then if there are leaves in , these leaves are all in the same subtree of . Thus, we have or .
Case 2.1. . Let be the only leaf in and be its neighbor. By remove this vertex, we get a tree with diameter . Suppose is a maximum linear forest in . Then is a linear forest in . Conversely, if is a maximum linear forest in then is a linear forest in . It follows that . Therefore, by adding one leaf to , we obtain a new tree with vertices and diameter as shown in Fig.4. And it is easy to see that for any in , . Thus,
[TABLE]
Case 2.2. . Let and be these two vertices and and be their neighbors, respectively. Clearly, . By remove these two vertices, we get a tree with diameter . Clearly, we have . Thus, by adding two leaves to , we obtain a tree with vertices and diameter as shown in Fig.5. And any tree T in with has . Thus,
[TABLE]
Since , then we have if and is the extremal tree that achieves the upper bound. if and is the extremal graph that achieves the upper bound.
Let be a tree on vertices. Then has edges. Since and . Then, we have the following corollary.
Corollary 3.2**.**
Let be a tree on vertices with diameter , then
[TABLE]
Theorem 3.2**.**
For any connected graph with vertices and edges, if the length of the longest path in is and , then we have
[TABLE]
*Proof. *According to Lemma 2.1, we have that and . Then it is clear that . Since any linear forest in can be extended to a spanning tree of , then there exists a spanning tree of such that . Moreover, the maximum linear forest in any spanning tree of is also a linear forest of . It implies that for any spanning tree of , . Consequently, let be the set of all spanning trees of , then . It is easy to see that the diameter of each spanning tree of is less than or equal to . Moreover, upper bounds on in Theorem 3.1 are all increasing functions of diameters. Thus,
[TABLE]
Thus, the theorem follows.
4 The Decycling Number of Line Graphs of -ary Trees
A -ary tree is a rooted tree where within each level every node has either [math] or children. The maximum degree of a -ary tree is . It follows that line graphs of -ary trees are -free graphs. Since it is often interesting to consider the decycling number of -free graphs. Thus, we consider the decycling number of line graphs of -ary trees in this section.
Before that, we give a dynamic programming algorithm to find a maximum linear forest in rooted tree. Let be a rooted tree with root . Let be subtrees of with root , respectively. Let be the edge set of a maximum linear forest in . Let be the edge set of a largest linear forest of such that the degree of is at most one.
Then, we define three kinds of linear forests , and as follows.
[TABLE]
As shown in Fig.6, is a linear forest in such that has degree zero; is a linear forest such that has degree one and is an edge in the linear forest; is a linear forest such that has degree two and , are edges in the linear forest. Let be the largest linear forest among all , and . Let be the largest linear forest among all , .
Lemma 4.1**.**
For any tree , is a maximum linear forest in , is a largest linear forest in such that root has degree at most one.
*Proof. *For any tree , if is not a maximum linear forest in . Let be a maximum linear forest in . Let , for . We can divide the proof into three cases according to the degree of in .
Case 1. The degree of in is zero. Clearly, is a linear forest in subtree . Then we have . Therefore, .
Case 2. The degree of in is one. Suppose is in . Then let , for . Clearly, for , is a linear forest in subtree and is a linear forest in subtree with degree of at most one. Thus, for each we have and . Therefore, .
Case 3. The degree of in is two. Suppose and are in . Then let . Clearly, for , is a linear forest in subtree . is a linear forest in subtree with degree of at most one and is a linear forest in subtree with degree of at most one. Thus, for each we have . For each , we have and . Therefore, .
Combining these cases, we get the conclusion that , which implies that is a maximum linear forest in . Similarly, we can prove is a largest linear forest in such that root has degree at most one.
Theorem 4.1**.**
For any -ary tree with vertices, we have
[TABLE]
*Proof. *For the lower bound, suppose has internal vertices and leaves, then we have . It follows that and . By Lemma 3.1, we have . Thus, .
For the upper bound, we can divide edges of into groups such that each edges with the same parent are in the same group. Since the degree of vertices in the linear forest is at most two. Thus, at most two edges in each group are in the linear forest. Therefore, we get .
Let be a -ary tree on vertices such that in each layer there is only one node with children, as shown in Fig. 7. Let be the internal vertex at depth and root is at depth [math]. For , it is clear that as shown in Fig. 7 (a). For , by Theorem 4.1 it is easy to check that the linear forest shown in Fig. 7 (b) is maximum. Thus, for is even; for is odd.
Corollary 4.1**.**
Let be a -ary tree on vertices, then
[TABLE]
A perfect -ary tree is a -ary tree in which all leaves are at the same depth. At last, we obtain the maximum linear forests in perfect -ary trees as follows.
Theorem 4.2**.**
For any perfect -ary tree with vertices, we have
[TABLE]
where is the height of and leaves are at height 1.
*Proof. *Let be the height of . We construct a linear forest of as follows. Firstly, we choose two vertex-disjoint paths of length that go from root to two leaves. Then tree is decomposed into subtrees of height , subtrees of height , subtrees of height , …, subtrees of height and subtrees of height as shown in Fig. 8. Then for each subtree, we choose two vertex-disjoint paths that go from the root to the two leaves again. Do it recursively, then a linear forest of is created.
Let be the edge set of the obtained linear forest in with height and let be the cardinality of . Then it is easy to see that and . According to the recursive construction of the obtained linear forest, we have
[TABLE]
and
[TABLE]
Combining the two equations, we get a recursive relation as follows.
[TABLE]
By the technique of generating functions, we can derive a formula for as follows.
[TABLE]
Thus, .
Now we prove that is a maximum linear forest in . Let be the root of and , , …, be the subtrees of with root . Since is a perfect -ary tree, each subtree is identical to a perfect -ary tree of height . Suppose and are in . Let be the subset of in subtree and let be the cardinality of . Then is a linear forest in such that .
We claim that in a perfect -ary tree of height , is a maximum linear forest and is a largest linear forest such that the degree of the root is at most one. We prove the claim by induction on . For and , it is easy to check , are maximum linear forests and , are largest linear forests with degree of root at most one, where are empty sets. Suppose the claim is true for perfect -ary tree with height . Let be a perfect -ary tree with height . Define
[TABLE]
By induction hypothesis, each is identical to and each is identical to . Then
[TABLE]
It is easy to see that each has vertices. Since is also consist of ’s, two ’s and two extra edges, then we have . It follows that . Similarity, we have . Then,
[TABLE]
It follows that . By Theorem 4.1, we see that is a maximum linear forest in and is a largest linear forest such that degree of the root is at most one, which are exactly and . Therefore, we prove the claim and is a maximum linear forest of . Thus, we conclude that .
Corollary 4.2**.**
For any perfect -ary tree with vertices, the decycling number of is
[TABLE]
where is the height of .
Acknowledgements.
The work was supported by National Natural Science Foundation of China (No.11671299, No.61502330, No.61472465, No.61562066).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] V. Bafna, P. Berman and T. Fujito, A 2-approximation algorithm for the undirected feedback vertex set problem, SIAM J. Discrete Math . 12 (1999), 289–297.
- 2[2] L. W. Beineke and R. C. Vandell, Decycling graphs, J. Graph Theory 25 (1997), 59–77.
- 3[3] P. Erdős, M. Saks and V. T. Sós, Maximum induced trees in graphs, J. Combin. Theory Ser. B 41 (1986), 61–79.
- 4[4] R. Focardi, F. L. Luccio and D. Peleg, Feedback vertex set in hypercubes, Inform. Process. Lett . 76 (2000), 1–5.
- 5[5] J. Fox, P. S. Loh and B. Sudakov, Large induced trees in K r subscript 𝐾 𝑟 K_{r} -free graphs. J. Combin. Theory Ser. B , 99(2) (2009), 494–501.
- 6[6] M. R. Garey and D. S. Johnson, Computers and Intractability, A guide to the theory of NP-completeness. A Series of Books in the Mathematical Sciences. W. H. Freeman and Co. , 1979. x+338 pp. ISBN: 0-7167-1045-5.
- 7[7] L. Gao, X. Xu, J. Wang et. al. The decycling number of generalized Petersen graphs, Discrete Applied Mathematics . 181 (2015), 297–300.
- 8[8] S. E. Goodman and S. T. Hedetniemi, On the Hamiltonian completion problem. Graphs and Combinatorics. Springer Berlin Heidelberg , (1974), 262–272.
