Edge separators for graphs excluding a minor

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sykora, and Vr\v{t}o (1993) for planar graphs, and of Sykora and Vr\v{t}o (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H \boxtimes K_{\lfloor p \rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = \sqrt{(t-3)\Delta |E(G)|} + \Delta$.


Introduction
A balanced vertex separator of an n-vertex graph G is a set X ⊆ V (G) such that every component of G − X has at most n/2 vertices. 1The well-known Planar Separator Theorem by Lipton and Tarjan [7] states that every n-vertex planar graph has a balanced vertex separator of size O( √ n).Alon, Seymour and Thomas [1] showed that every n-vertex K tminor-free graph has a balanced vertex separator of size at most t 3/2 √ n.
In this paper, we study balanced edge separators.A balanced edge separator of an n-vertex graph G is a set F ⊆ E(G) such that every component of G − F has at most n/2 vertices.The aforementioned classes of graphs with balanced vertex separators of size O( √ n) do not admit balanced edge separators of size o(n); indeed, the smallest balanced edge separator of the n-vertex star K 1,n−1 has size ⌈n/2⌉.
The star, however, has a vertex of degree n − 1.If we assume that the maximum degree ∆ of G is sublinear in n, then in some cases we can retrieve sublinear balanced edge separators.Diks, Djidjev, Sỳkora, and Vrťo [3] showed that if G is planar, then G has a balanced edge separator of size O( √ ∆n), and Sỳkora and Vrťo [8] showed that if the Euler genus of G is g, then there exists a balanced edge separator of size O( √ g∆n).These results are true also in the weighted setting where each vertex x ∈ V (G) is assigned a weight w(x) with 0 w(x) 1  2 , the total weight of all vertices is 1, and the edge separator should split G into components of weight at most 1  2 .Lasoń and Sulkowska [6] showed that in the weighted setting, if G is an n-vertex K t -minorfree graph of maximum degree ∆ = o(n) and the vertices are weighted proportionally to their degrees, then there exists a balanced edge separator of size o(n).Their proof relies on spectral methods-more precisely, on an upper bound for the second smallest eigenvalue of the Laplacian matrix of K t -minor-free graphs due to Biswal, Lee, and Rao [2]-and only works for these specific weights.They asked if one can always find a balanced edge separator of size O t ( √ ∆n) for any weights (including uniform weights), as is the case for planar graphs and graphs of bounded genus.In this paper, we give an affirmative answer to this question.
Theorem 1.Let t 3, Let G be an n-vertex K t -minor-free graph of maximum degree ∆, and let w : V (G) → [0, 1  2 ] be a weight function such that x∈V (G) w(x) = 1.Then there exists a set This result is best possible up to dependency on t: Sỳkora and Vrťo [8] showed that there exist n-vertex planar graphs G of maximum degree ∆ such that every balanced edge separator has size Ω( √ ∆n).
We actually prove the following stronger result.
Theorem 2. Let t 3 and let G be a K t -minor-free graph of maximum degree ∆ with m edges.Then the line graph of G is isomorphic to a subgraph of the strong product H ⊠ K ⌊p⌋ for some graph H with tw(H) t − 2 and p = (t − 3)∆m + ∆.
The strong product H ⊠ K of graphs H and K is a graph on V (H) × V (K) where (x 1 , y 1 ) and (x 2 , y 2 ) are adjacent if x 1 = x 2 and y 1 y 2 ∈ E(K), or x 1 x 2 ∈ E(H) and y 1 = y 2 , or x 1 x 2 ∈ E(H) and y 1 y 2 ∈ E(K).(When K is a complete graph on p vertices, taking the strong product of H with K amounts to 'blowing up' each vertex of H by a clique of size p.) Theorem 2 directly implies the following upper bound on the treewidth of the line graph of G.
Theorem 3. Let t 3, and let G be a K t -minor free graph of maximum degree ∆ with m edges.Then the line graph of G has treewidth at most (t − 1)⌊ (t − 3)∆m + ∆⌋ − 1.
Theorem 1 then follows from Theorem 3 by a simple argument on the tree-decomposition provided by the latter theorem (see Section 3).Theorem 2 can be thought of as an 'edge version' of the following recent strengthening of the balanced vertex separator result by Alon, Seymour and Thomas [1] due to Illingworth, Scott and Wood [4]: Every n-vertex K t -minor free is isomorphic to a subgraph of the strong product H ⊠ K ⌊p⌋ where tw(H) t − 2 and p = (t − 3)n.The authors of [4] established their result by modifying the proof in [1].Our proof of Theorem 2 is likewise a modification of the proof in [1] and relies heavily on the insights from [4]; the main work consists in adapting to the edge setting.
One consequence of Theorem 1 is an upper bound for the isoperimetric number (a.k.a.edge expansion or Cheeger constant) of K t -minor-free graphs.The isoperimetric number φ(G) of a graph G is defined as Corollary 4. For t 3, every n-vertex K t -minor-free graph G with maximum degree ∆ satisfies given by Theorem 1.Since each component of G − F has at most n/2 vertices, we may choose a subset of these components so that the union S of their vertex sets satisfies The bound in Corollary 4 is best possible up to the dependence on t, and extends previous bounds for planar graphs [3] and bounded-genus graphs [8].
Our proofs are constructive, and in particular, there exists a polynomial time algorithm, which given a graph G and an integer t, outputs a set F as in Theorem 1 and a graph H as in Theorem 2 together with an isomorphism between the line graph of G and a subgraph of H ⊠ K ⌊p⌋ where p = (t − 3)∆m + ∆.
In Section 2 we introduce all the necessary definitions, notations and preliminary results, and in Section 3 we prove Theorems 1, 2 and 3.

Preliminaries
We consider simple finite undirected graphs G with vertex set V (G) and edge set E(G).For subsets X, Y ⊆ V (G), we denote by E G (X, Y ) the set of all edges xy ∈ E(G) with x ∈ X and y ∈ Y .We denote by N G (X) the open neighborhood of a set X ⊆ V (G), i.e. the set of all vertices outside X that are adjacent to at least one vertex from X.We drop the subscripts from the notations E G (X, Y ) and N G (X) when the graph G is clear from the context.
A tree-decomposition of a graph G is a family (B u ) u∈V (T ) of subsets of V (G) called bags, indexed by nodes of a tree T , such that • for every xy ∈ E(G) there exists u ∈ V (T ) with {x, y} ⊆ B u , and • For all u 1 , u 2 , u 3 ∈ V (T ) such that u 2 lies on the path between u 1 and u 3 in T , we have The width of The treewidth of a graph G, denoted tw(G), is the minimum width of its tree-decompositions.The following fact summarizes some simple properties of tree-decompositions that we use in our proof.
Fact 5. Given a graph G with a tree-decomposition (B u ) u∈V (T ) of width at most k, the following properties hold: (ii) Every graph G ′ obtained from G by adding a new vertex adjacent to a clique of G of size at most k has treewidth at most k.(iii) For every clique X in G there exists u ∈ V (T ) with X ⊆ B u .(iv) For every connected set U ⊆ V (G), the set {u ∈ V (T ) : The line graph L(G) of a graph G is a graph whose vertex set is E(G) and in which distinct edges e, e ′ ∈ E(G) are adjacent if they share a common end in G.
Given a graph G, a graph-partition of G is a graph H such that the vertex set of H is a partition of V (G) into nonempty parts, and for all distinct X, Y ∈ V (H) we have

The proofs
We need the following lemma by Alon, Seymour and Thomas [1].

Lemma 6 ([1]
).Let G be a graph, let A 1 , . . ., A h be h subsets of V (G), and let r be a real number with r 1.Then either: • there is a tree We deduce an edge variant of this lemma.
Lemma 7. Let G be a graph without isolated vertices, let A 1 , . . ., A h be h subsets of V (G), and let r be a real number with r 1.Then either: • there is a tree T in G with |E(T )| r such that V (T )∩A i = ∅ for every i ∈ {1, . . ., h}, or • there exists G)) = ∅ for every i ∈ {1, . . ., h}, then G contains a tree T such that E(T ) = V (T 0 ), and thus |E(T )| = |V (T 0 )| r and V (T ) ∩ A i = ∅ for every i ∈ {1, . . ., h}.

Now suppose that
intersects all of A 1 , . . ., A h , then F satisfies the lemma, and we are done.Assume thus that some component C of G − F intersects all of A 1 , . . ., A h .By our assumption on F , there exists i ∈ {1, . . ., h} such that E(C) ∩ E G (A i , V (G)) = ∅.Hence, C must consist of a single vertex that belongs to all of A 1 , . . ., A h , and therefore the tree T := C satisfies the lemma.
The following lemma is the heart of the proofs of our results.Lemma 8. Let t, ∆, m, h be integers with t 3 and ∆, m, h 1, and let p := (t − 3)∆m + ∆.Let G be a connected K t -minor-free graph of maximum degree at most ∆ with m edges, let C be a proper induced subgraph of G with |V (C)| 1, and let E 1 , . . ., E h be disjoint nonempty Proof.We prove the lemma by induction on the value for each i ∈ {1, . . ., h}, so we may apply the induction hypothesis to C α to obtain a . Therefore, we may assume that C is connected.
For each i ∈ {1, . . ., h}, let A i = V (C) ∩ N (U i ).Suppose that some A i is empty, say without loss of generality A h = ∅.Since G is connected, not all sets A i are empty, so h 2. We have so we may apply the induction hypothesis to C and the sets edge of E h is incident to an edge in E(C), and thus the line graph L(G) does not contain any edges between E h and E(C).Hence, by Fact 5(ii), the desired can be obtained from H 0 by adding E h as a new vertex adjacent to E 1 , . . ., E h−1 .Therefore, we may assume that the sets A 1 , . . ., A h are nonempty.
Suppose that C contains a tree T on at most (h − 1)m/∆ + 1 vertices that contains at least one vertex in A i for each i ∈ {1, . . ., h}.Let U h+1 = V (T ), so that (U 1 , . . ., U h+1 ) is a K h+1 -model, and let |+h, so we may apply the induction hypothesis to C − V (T ), (U 1 , . . ., U h+1 ) and the sets E 1 , . . ., By the minimality of F , each edge e ∈ F with an end in C j belongs to one component of C − (F \ {e}) with an element of A ′ i , so e has an end in Therefore, we may apply the induction hypothesis to C j and the sets E rooted at {F, E 1 , . . ., E h } and in particular at {E 1 , . . ., E h }.This concludes the proof of the lemma.
Proof of Theorem 2. Let G be a K t -minor-free graph of maximum degree ∆ with m edges, and let G 1 , . . ., G s be the components of G.For each j ∈ {1, . . ., s}, we construct a (t − 2, p)partition H j of L(G j ).If G j is an isolated vertex, then L(G j ) is an empty graph and we can take the empty graph as H j .If G j is not an isolated vertex, then choose any vertex x ∈ V (G j ).By Lemma 8 applied to C = G j − x, E 1 = E({x}, V (G)) and U 1 = {x}, L(G j ) has a (t − 2, p)-partition H j .Hence, by Fact 5(i), H = H 1 ∪ • • • ∪ H s is a (t − 2, p)-partition of L(G), and therefore L(G) is isomorphic to a subgraph of H ⊠ K ⌊p⌋ .
Proof of Theorem 1.Let G be a K t -minor-free graph of maximum degree ∆ with n vertices and m edges, and let w : V (G) → [0, 1  2 ] be a weight function such that x∈V (G) w(x) = 1.
there is no restriction in this case.)For an integer k and a real p, we call a graph-partition H of G a (k, p)-partition if k and |X| p for each X ∈ V (H).Observe that if a graph G has a (k, p)-partition then G is isomorphic to a subgraph of H ⊠ K ⌊p⌋ for some graph H with tw(H) k.In the proofs, we will often consider a graph-partition H of a graph G together with some distinguished clique {X 1 , . . ., X h } of H; in this case, we say that H is rooted at {X 1 , . . ., X h }.
at {E 1 , ..., E h }, as desired.It remains to consider the case when no tree in C with at most (h − 1)m/∆ + 1 vertices intersects all of A 1 , ..., A h .In particular, h 2. Since C is connected and |V (C)| > 1, C does not contain isolated vertices.By Lemma 7 with r = (h − 1)m/∆, there exists a set such that no component of C − F intersects all of A 1 , ..., A h .Among all such sets F , choose a smallest one.Since C is connected and the sets A 1 , ..., A h are nonempty, F = ∅.Let C 1 , ..., C s be the components of C − F .Observe that C 1 , ..., C s are induced subgraphs of C (and thus of G), by our choice of F .Our goal now is to show that for each j ∈ {1, ..., s}, there exists a (t−2, p)-partitionH j of L(G)[E(C j )∪F ∪E 1 ∪• • •∪E h ] rooted at {F, E 1 , . .., E h }.By Fact 5(i), this will then imply that L(G)[E(C) ∪ E 1 ∪ • • • ∪ E h ] admits a (t − 2, p)-partition H rooted at {E 1 , . .., E h }, as desired.Towards this goal, fix j ∈ {1, . . ., s}, and let i ′ ∈ {1, . . ., h} be such that V (C j ) ∩ A i ′ = ∅.Consider the sets E ′ 1 , . . ., E ′ h where E