The treewidth of 2-section of hypergraphs

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.


Introduction
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. The concept of treewidth was originally introduced by Bertelé and Brioschi [2] under the name of dimension. It was later rediscovered by Halin [7] in 1976 and by Robertson and Seymour [13] in 1984. Now it has been studied by many other authors (see for example [5]- [12]). Treewidth is commonly used as a parameter in the parameterized complexity analysis of graph algorithms, since many NP-complete problems can be solved in polynomial time on graphs of bounded treewidth [3]. The relation between the treewidth and other graph parameters has been explored in a number of papers (see [10] for a recent survey). In [11], Harvey and Wood studied the treewidth of line graphs. They proved sharp lower bounds of the treewidth of the line graph of a graph G in terms of both the minimum degree and the average degree of G. Motivated by their work, in this paper, we study the treewidth of 2-section of linear hypergraphs.
A hypergraph is a pair H = (V, F ), where V is a finite set of vertices and F is a family of subsets of V such that for any f ∈ F , f = ∅ and V = ∪ f ∈F f . The size of H is the cardinality of F . A simple hypergraph is a hypergraph H such that if f ⊆ g, then f = g, where f, g ∈ F . If |f | = 1, we call f a loop. In this paper, we just consider simple and no loop hypergraphs. The rank and anti-rank of H is defined as r(H) = max Proof: Let H = (V, F ) be a 2-regular linear hypergraph. Then H * is a simple graph. By the definition of dual hypergraph, there is a bijection σ between the edge set F (H) (resp. the vertex set V (H)) and the vertex set V (H * ) (resp. the edge set of E(H * )) such that for any f, g ∈ F (H) (resp. for any u, v ∈ V (H)), σ(f )σ(g) ∈ E(H * ) if and only if f ∩ g = ∅ (resp. σ(u) ∩ σ(v) = ∅ if and only if there is f ∈ F (H) with u, v ∈ f ). By the definition of line graph, there is a bijection φ : E(H * ) → V (L(H * )) such that for any e 1 , e 2 ∈ E(H * ), φ(e 1 )φ(e 2 ) ∈ E(L(H * )) if and only if e 1 ∩ e 2 = ∅.
We will show that We prove the following lower bound on tw([H] 2 ) in terms of the minimum degree δ, maximum degree ∆ and average rank l(H) of a linear hypergraph H. Theorem 1.1 Let H be a linear hypergraph with minimum degree δ, maximum degree ∆ and average rank l(H).
In [11], Harvey and Wood showed that for every graph G, tw(L(G)) > 1 is the average degree of G. Let H be a 2-regular linear hypergraph of order n and size m. By Lemma just as the result in [11].
We also prove two lower bounds on tw([H] 2 ) in terms of the anti-rank s(H) based on different condition of minimum degree of the given hypergraph H. when s(H) is odd.
In [11], Harvey and Wood showed that for every graph G with minimum degree δ(G), To see this, we consider a minimum width supertree decomposition of H, and replace each bag λ t by the vertices that are incident to an hyperedge of λ t . This creates a tree decomposition of [H] 2 , where each bag contains at most r(H)stw(H) vertices. In Section 5, we improve this bound as follows.  In [11], Harvey and Wood showed that for every graph G, tw(L(G)) ≤ 2 3 (tw(G) + 1)∆(G) + just the same as that in [11].
The rest of this paper is organized as follows. In Section 2, some properties of tree decompositions of 2-section of hypergraphs are given. The proof of Theorem 1.1 is given in Section 3. In Section 4, we will prove Theorems 1.2 and 1.3. In Section 5, we prove Theorem 1.4. Section 6 concludes this paper.

Tree decomposition of 2-section of hypergraphs
In this section, we first give some properties of tree decompositions of 2-section of hypergraphs which will be used in next sections. Let For u, v ∈ T , we use P ath(u, v) to denote the path in T connecting u and v.
Thus there exists a bag B t containing all the vertices in f , where t ∈ T . Hence for each f ∈ F choose one such node and declare it b(f ). Let , the removal of the aforementioned vertices yields another tree decomposition of [H] 2 . However, the existence of such a tree decomposition would contradict our choice of (T, We call b(f ) the base node of f . From the proof of Lemma 2.1, we can construct a tree decomposition of [H] 2 so that we can assign a base node for each f ∈ F and all the vertices in f are placed in the corresponding bag B b(f ) . In fact, we can obtain a slightly stronger result that will be used to prove our theorems.
Given (T, (B t ) t∈T ) and b guaranteed by Lemma 2.1, we can also ensure that each base node is a leaf and that b is a bijection between edges of H and leaves of T . If b(f ) is not a leaf, then we add a leaf adjacent to b(f ) and let b(f ) be this leaf instead. If some leaf t is the base node for several edges of H, then we add a leaf adjacent to t for each edge assigned to t. Finally, if t is a leaf that is not a base node, then delete t; this maintains the desired properties since a leaf is never an internal node of a subtree.
We can improve this further. Given a tree T , we can root it at a node and orient all edges away from the root. In such a tree, a leaf is a node with outdegree 0. Say a rooted tree is binary if every non-leaf node has outdegree 2. Given a tree decomposition, by the same way described in [11], we can root it and then modify the underlying tree so that each non-leaf node has outdegree 2. The above results give the following lemma.
By Lemma 2.2, we have the following lower bound on tw([H] 2 ) that is slightly stronger than (2).
. Then for all t, |λ t | ≤ k(∆ − 1) since each vertex contributes at most ∆ − 1 hyperedges to a given bag. An Repeat this process so that every edge in T ′ corresponds to at most one 3-tuple. We also use T ′ to denote the tree after all of these subdivisions. Note that We are going to show that First, we prove that for all t ∈ T ′ , B ′ t ⊆ ∪ f ∈λt f . If t ∈ T , then the conclusion holds because for each v ∈ B t , we put at least one edge incident to v in λ t . If t is the subdivided node, then the result holds by the construction.
, then we should have f ∈ λ t ′′ for all t ′′ ∈ P ath(t, t ′ ) even if t ′′ is a subdivided node by the construction, a contradiction.
Firstly, we need some lemmas before we start bounding the tree width of [H] 2 .
Lemma 3.1 If H is a minimal hypergraph and S is a nonempty proper subset of F (H), then and we have the conclusion.
Proof: Denote So g(z(T a ), z(T b )) > 1 ∆ (|z(T a )| + |z(T b )|)l(H). We consider B t , the bag of the target node t, which consists of vertices covered by edges in z(T a ) (resp. z(T b )) and F \ z(T a ) (resp. F \ z(T b )) at the same time. Thus, So the above formula can be written as thus we have the conclusion.
Let (T, (B t ) t∈T ) be a tree decomposition of [H] 2 as guaranteed by Lemma 2.2. Call a node t of T significant if |z(T t )| > l(H) 2 but |z(T t ′ )| ≤ l(H) 2 for each child t ′ of t. Claim 1. There exists a non-root, non-leaf significant node t.
Proof of Claim 1: Starting at the root of T , begin traversing down the tree by the following rule: if some child t of the current node has |z(T t )| > l(H) 2 , then traverse to t; otherwise stop. Clearly this algorithm halts.
Since |z(T t )| = 1 ≤ l(H) 2 for any leaf t, the algorithm will stop at a non-leaf. Let t be the node where the above algorithm stops. Suppose that t is the root. Then |z(T t )| = |F |.
We consider the following linear programming.
where the constraint condition is based on H being a linear hypergraph, that is, each pair (f i , f j ) can only be calculated at most one in σ i i (A). From the above linear programming, we can easily know that Define α, β, s such that |A| = αl(H), |B| = βl(H) and s = 1 l(H) . Recall |A|, |B| ≤ 1 2 l(H) and |A| + |B| > 1 2 l(H). Hence |A|, |B| ≥ 1. Thus s ≤ α, β ≤ 1 2 and α + β > 1 2 . Now we have In Appendix A, we prove that f (α, Let k ≥ 2 be a positive integer. We construct a hypergraph H = (V, F ) where F = {f 1 , f 2 , . . . , f n }. The vertex set of H is determined as the following rule: for any positive integers i, j with i = j, if |i − j| ≤ k, then we put the vertex v i,j into both f i and f j , where we let v i,j = v j,i . Then V = ∪ n i=1 f i . We can easily know that H is a 2-regular linear hypergraph with l(H) = 2k − γ, where γ → 0 when n → ∞. By Corollary 3.4, tw( where P k n is the k th -power of an n-vertex path with vertex set {v 1 , v 2 , . . . , v n } and edge set {(v i , v j ) | |i − j| ≤ k, 1 ≤ i < j ≤ n}. In [11], Harvey and Wood showed that tw(L(P k n )) ≤ 1 2 k 2 + 3 2 k − 1. Thus when h = 2, Corollary 3.4 is almost precisely sharp for treewidth. We first give the proof of Theorem 1.2 where δ ≥ 3. As in Lemma 3.2,

Lower bounds in terms of anti-rank
Thus we have Consider the partial derivative of f (α, β) with respect to α and β, we have In Appendix C, we show that Thus When s(H) is even, let H = (C k n ) * , where C k n is the k th -power of an n-vertex cycle with vertex set {v 1 , v 2 , . . . , v n } and edge set {(v i , v j ) | min{|i − j|, i + n − j} ≤ k, 1 ≤ i < j ≤ n}. We can see that in this case H is a 2-regular linear hypergraph and s(H) = δ(C k n ) = 2k. By Theorem 1.
Thus when s(H) is odd, Theorem 1.3 is precisely sharp when n is even; and within '+1' when n is odd.

Upper bound
Proof of Theorem 1.4: Let (T, (B t ) t∈T , (λ t ) t∈T ) be a supertree decomposition of a linear hypergraph H with width k such that T has maximum degree at most 3. By the discussion in Section 1, we may assume that r(H) ≥ k − 1. (The existence of such a supertree decomposition (T, (B t ) t∈T , (λ t ) t∈T ) is well known and follows by a similar argument to Lemma 2.2.) Say a hyperedge f ∈ F is small if |f | ≤ k − 1 and large otherwise. For each f ∈ F , we use T (f ) to denote the subtree of T induced by λ −1 (f ). For each edge e in T , let A(e), B(e) denote the two subtrees of T − e. If e is also an edge of T (f ) for some f ∈ F (H), then let A(e, f ), B(e, f ) denote two subtrees of T (f ) − e, where A(e, f ) ⊆ A(e) and B(e, f ) ⊆ B(e). For t ∈ λ −1 (f ), let γ t (f ) = {v ∈ V (H)|v ∈ f ∩ g, g ∈ λ t \ {f }}. Since H is a linear hypergraph, we have |γ t (f )| ≤ k − 1. Denote α(e, f ) = ∪ t∈A(e,f ) γ t (f ) and β(e, f ) = ∪ t∈B(e,f ) γ t (f ). We have the following claim.

Claim 2.
For every large f ∈ F there is an edge e in T (f ) such that |α(e, f )|, |β(e, f )| ≤ 2 3 |f |+ 1 3 (k −1). Proof of Claim 2: Assume for the sake of a contradiction that no such e exists. Hence for all e in T (f ), either |α(e, f )| or |β(e, f )| is too "large". Direct the edge e towards A(e, f ) or B(e, f ) respectively. (If both |α(e, f )|, |β(e, f )| are too large, then direct e arbitrarily.) Given this orientation of T (f ), there must be a sink (all the edges incident to it direct to it), which we label t 0 .