A lower bound on the tree-width of graphs with irrelevant vertices

For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function f such that if the tree-width of a graph G with k pairs of terminals is at least f(k), then G contains a solution-irrelevant vertex (Graph Minors. XXII., JCTB 2012). We give a single-exponential lower bound on f. This bound even holds for planar graphs.

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1 Introduction

The Disjoint Paths Problem is one of the famous classical problems in the area of graph algorithms. Given a graph , and pairs of terminals, , it asks whether contains vertex-disjoint paths such that connects to , (for ). Karp proved that the problem is NP-hard in general Karp75 and Lynch proved that it remains NP-hard on planar graphs Lynch75 . Robertson and Seymour showed that it can be solved in time for some computable function , i. e. the problem is fixed-parameter tractable (and, in particular, solvable in polynomial time for fixed ). For a recursive step in their algorithm ((10.5) in GMXIII ), they prove GMXXII that there is a function such that if a graph with pairs of terminals has tree-width at least , then contains a vertex that is irrelevant to the solution, i. e.  contains a non-terminal vertex such that has a solution if and only if the graph (with the same terminals) has a solution.

In this paper we give a lower bound on , showing that , even for planar graphs. For this we construct a family of planar input graphs , each with pairs of terminals, such that the tree-width of is , and every member of the family has a unique solution to the Disjoint Paths Problem, where the paths of the solution use all vertices of the graph. Hence no vertex of is irrelevant. As a corollary, we obtain a lower bound of on the tree-width of graphs having vital linkages (also called unique linkagesGMXXI with components.222This result appeared in the last section of a conference paper AdlerKKLST11 . While the main focus of the paper AdlerKKLST11 was a single exponential upper bound on on planar graphs, it only sketches the lower bound. Here we provide the full proof of the lower bound. A longer proof of the lower bound can also be found in the thesis Krause2016 . Our result contrasts the polynomial upper bound in a related topological setting Matousek2016 , where two systems of curves are untangled on a sphere with holes.

For planar graphs, an upper bound of was given in AdlerKKLST11 . An elementary proof for a bound of was provided later Krause2016 as well as a slightly improved bound of requiring a slightly more involved proof AdlerKKLST17 . Our lower bound shows that this is asymptotically optimal. Recently, an explicit upper bound on on graphs of bounded genus GeelenHR18 was found, then refined into one that is single exponential in and the genus Mazoit13 . The exact order of growth of on general graphs is still unknown.

2 Preliminaries

Let denote the set of all non-negative integers. For , we let . For a set we let denote the power set of . A graph is a pair of a set of vertices and a set of edges , i. e. graphs are undirected and simple. For an edge , the vertices and are called endpoints of the edge , and the edge is said to be between its endpoints. For a graph let and . Let and be graphs. The graph is a subgraph of (denoted by ), if and . For a set , the subgraph of induced by is the graph and we let .

A path in a graph is a sequence of pairwise distinct vertices of , such that for every there is an edge . The vertices and are called endpoints of . The path is called a path from to (i. e. paths are simple). We sometimes identify the path in with the subgraph of . A graph is called connected, if it has at least one vertex and for any two vertices , there is a path from to in . The inclusion-maximal connected subgraphs of a graph are called connected components of the graph. For , a set separates from , if there is no path from a vertex in to a vertex in in the subgraph of induced by . A tree is a non-empty graph , such that for any two vertices there is exactly one path from to in .

Figure 1: -grid

Let . The )-grid is a graph with and . In case of a square grid where , we say that is the size of the grid. An edge in the grid is called horizontal, if , and vertical, if . See Figure 1 for the -grid.

A drawing of a graph is a representation of in the Euclidean plane , where vertices are represented by distinct points of and edges by simple curves joining the points that correspond to their endpoints, such that the interior of every curve representing an edge does not contain points representing vertices. A planar drawing (or embedding) is a drawing, where the interiors of any two curves representing distinct edges of are disjoint. A graph is planar, if has a planar drawing (See MoharTh01 for more details on planar graphs). A plane graph is a planar graph together with a fixed embedding of in . We will identify a plane graph with its image in . Once we have fixed the embedding, we will also identify a planar graph with its image in .

Definition 1 (Disjoint Paths Problem ()).

Given a graph and pairs of terminals , the Disjoint Paths Problem is the problem of deciding whether contains vertex-disjoint paths such that connects to (for ). If such paths exist, we refer to them as a solution. We denote an instance of by .

Let be an instance of . A non-terminal vertex is irrelevant, if has a solution if and only if has a solution.

A tree-decomposition of a graph is a pair , consisting of a tree and a mapping , such that for each there exists with , for each edge there exists a vertex with , and for each the set is connected in . The width of a tree-decomposition is

If is a path, is also called a path-decomposition. The tree-width of is

The path-width of is

Obviously, every graph satisfies . Every tree has tree-width at most and every path has path-width at most . It is well known that the )-grid has both tree-width and path-width . Moreover, if , then and .

Theorem 1 (Robertson and Seymour Gmxxii ).

There is a function such that if , then has an irrelevant vertex (for any choice of terminals in ).

A linkage in a graph is a subgraph , such that each connected component of is a path. The endpoints of a linkage are the endpoints of these paths, and the pattern of is the matching on the endpoints induced by the paths, i. e. the pattern is the set

A linkage in a graph is a vital linkage in , if and there is no other linkage in with the same pattern as .

Theorem 2 (Robertson and Seymour Gmxxii ).

There are functions such that if a graph has a vital linkage with components then and .

3 The lower bound

Our main result is the following.

Theorem 3.

Let be as in Theorems 1 and 2. Then , , and . Moreover, this holds even if we consider planar graphs only.

Figure 2: with solution.

In our proof we construct a family of graphs , of tree-width and path-width , and with a vital linkage with components. Figure 2 shows the graph .

Figure 3: The construction of from .
Definition 2 (The graph ).

Let . We inductively define an instance of as follows.

The Graph is the path with vertices, , . The bottom row and the top row of are the graph itself.

We define the graph by adding a path with vertices to as follows. Let be the bottom row of and let be the top row of . Let

We set , and , for . The top row of is and the bottom row of is .

Let . We define the instance as .

Figure 3 shows the construction of from .

Remark 1.

By construction, the graph contains a -grid as a subgraph. The tree-width and path-width of are thus at least .

Remark 2.

By construction, the graph contains a linkage (because in each step we add a path linking a new terminal pair).

We will now show that this linkage is vital by considering a topological version.

Definition 3 (Topological DPP).

Given a subset of the plane and pairs of terminals the topological Disjoint Paths Problem is the problem of deciding whether there are pairwise disjoint curves in , such that each curve is homeomorphic to and its ends are and . If such curves exist, we refer to them as a solution. We denote an instance of the topological Disjoint Paths Problem by .

A disc-with-edges is a subset of the plane containing a closed disc such that the connected components of , called edges, are homeomorphic to open intervals . We now define a family of discs-with-edges together with terminals. These will be used as instances of the topological . Figure 4 illustrates the construction.

Figure 4: The construction of , for the topological . Note that and are only used to place correctly.
Definition 4 ().

Let be a closed disc in the plane and . We start by inductively defining points , on the boundary of . (These will be used as terminals and to confine the way the edges are added to .) Let be two distinct points on , and let . Hence is the union of two curves, each homeomorphic to the open interval . Call one of the curves and the other . Assume that , , , , and are already defined, and assume that is a curve adjacent to and . Place a new point on and a new point on , let , let be the component of adjacent to and , and let be the component of adjacent to and .

Now let and . Assume the space and the set are already defined. We define by adding a planar matching of edges to . We call the set of these edges . The edges are pairwise disjoint and disjoint from . They are added such that each end is adjacent to a point on and no two edges are adjacent to the same point on . Each edge has one end adjacent to a point on the component of between and , and the other end adjacent to a point on the component of between and . Finally, let .

In this way we obtain a family , of instances to the topological DPP.

Figure 5: A solution of the topological from Figure 4.
Remark 3.

The embedding of (as shown in Figure 2 for ) corresponds to the space . Thus by Remark 2 the topological on , has a solution.

For an instance of the topological on , this solution can be seen in Figure 5.

Lemma 1.

For the topological instance has a unique solution (up to homeomorphism). The solution uses all edges .

Figure 6: The solution on induced by the solution on .
Proof.

For this is true because . Inductively assume that the lemma holds for . Let be any solution to , . This solution induces a solution of the topological , , as follows. Every edge together with the segment of that connects the ends of and contains bounds a disc . The space is homeomorphic to and the paths form a solution of , , . Figure 6 illustrates this for . By induction, this solution is unique up to homeomorphism and the paths use all edges in . Let be the solution obtained by embedding the graph (cf. Remark 3). By uniqueness, for each , the edges of used by are the same as for , and the order of their appearance on when walking from to is also the same as on . Hence the solution on restricted to the closed disc of is a planar matching of curves (the curves in ) between pairs of points on (and the same pairs of points are obtained by restricting to ). These pairs of points also have to be matched in .

We now claim that in the solution on , each curve in uses an edge of . If not, then there is a curve

that avoids all edges in . Since the edges of are already used, is routed within . By construction of and the fact that all edges of are already used, this means that separates from both and the endpoints of the edges in , a contradiction to being a path in the solution. Hence uses an edge of .

Since the sets and have equal size, it follows that each curve of the matching

uses precisely one edge of . Since the endpoints of the matching are fixed, they induce an order on the matching curves which determines precisely which edge of is used by which curve.

Altogether, this shows that the solution to is unique up to homeomorphism and uses all edges .

Figure 7: The number of edges around the terminals is crucial (cf. Remark 4).
Remark 4.

In a topological instance, the number of edges around the terminals is crucial. Even just relaxing the conditions on by having 2 edges instead of 1 edge around terminal allows a quite different solution to the topological . This solution uses no edge around , one edge around each of , and the two edges around (Figure 7 shows this for ).

Theorem 4.

Let . The graph contains a vital linkage.

Proof.

Let be the linkage from Remark 2. We argue that it is vital. For and , one can easily verify that has a unique embedding. For , contracting an edge at suffices to make -connected. Since 3-connected planar graphs have unique embeddings Whitney1932 , the graph also has a unique embedding, and it suffices to consider our previous embedding of (cf. Figure 2). Let be the minimal disc containing the grid in . The disc together with is the space . The paths thus give a solution to the topological instance , which by Lemma 1 is unique and uses all edges in . Thus any linkage with the same pattern as can differ from only inside the grid. Thus for each there is a subpath of some path of the solution , such that the endpoints of are and for some . Hence the family is a linkage between the first column and the last column of the grid.

Suppose that indeed differs from . Then at least one path contains a vertical edge in the grid. Hence the column of contains at most vertices that are not used by and, by Menger’s Theorem Menger1927 , the remaining paths of the family cannot be routed, a contradiction.

Proof of Theorem 3 Theorem 3 immediately follows from Theorem 4 and Remark 1. ∎

Acknowledgements

This research was partially supported by the Deutsche Forschungsgemeinschaft, project Graphstrukturtheorie und algorithmische Anwendungen, AD 411/1-1 and project Graphstrukturtheorie im Übersetzerbau, KR 4970/1-1. We thank Frédéric Mazoit for valuable discussions, especially for inspiring Remark 4. We also thank an anonymous reviewer for suggesting very elegant shortenings of our construction and proof.

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