Treewidth Bounds for Planar Graphs Using Three-Sided Brambles

The treewidth of a graph is a fundamental idea in Robertson and Seymour’s pioneering work on graph minors. For any graph $G$, let $TW(G)$ be the treewidth of $G$ (see [10]). The base case in the proof of the Graph Minor Theorem [12] relies on square grids for two reasons: (1) the $n\times n$ square grid has treewidth $n$, so the family of square grids has unbounded treewidth, and (2) each square grid is planar and hence has genus zero. Robertson and Seymour use their well-known Grid-Minor Theorem (also called the Excluded Grid Theorem) to start an induction on the genus of a graph.

We replace squares with triangles in this role. In a graph, every 4-cycle contains a 3-cycle as a minor. In this sense, searching for a triangle is more general than searching for a square.

We proceed by addressing the dual problem to treewidth, finding a graph’s bramble number. Let $BN(G)$ be the bramble number of a graph $G$. Seymour and Thomas first introduced brambles (originally called screens) in [13] to get lower bounds on the treewidth of a graph.

Our method for bounding treewidth will be to define a special class of brambles called nets. Nets were introduced in Smith’s thesis [14], where they were used to construct two families of planar graphs, each containing a minor minimal obstruction to any treewidth. Nets are three-sided brambles in plane graphs. A formal definition of nets appears in Section 3.

Nets can be thought of as a generalization of a natural bramble in a square grid called the bramble of crosses, described in [2]. Each cross contains vertices from one row and one column in the grid. Smith [14] defines $n$-triangular grids for $n\geq 3$ (see figure 2), and considers brambles whose elements meet all three sides of the triangle (rather than four sides, as in the bramble of crosses in the square grid). In particular, he proves that this bramble in the $n$-triangular grid has order $n$. Thus, triangular grids are a natural three-sided analogue to square grids. In this context, a bramble of crosses becomes a bramble of trees, each with a unique root and three branches. This three-sided bramble in a triangular grid provides a canonical example of a net in a plane graph.

It is important to note that the $n$-triangular grid does not contain an $n\times n$ square grid. The largest square grid minor has side-length less than or equal to $(n+1)/\sqrt{2}$ because, by counting vertices, there are $\left({}^{n+1}_{\;\;\;2}\right)$ vertices in an $n$-triangular grid and $m^{2}$ vertices in an $m\times m$ square grid. Thus, the bramble number of a triangular grid is larger than the bramble number of any of its square grid minors. These examples motivate us to improve lower bounds on treewidth for planar graphs found using square grids by finding high order three-sided nets.

We define $\lambda(G)$ to be the largest order of any net in a subgraph of $G$. Let $s(G)$ be the size of the side of the largest square grid minor in $G$. Smith [14] has shown that $s(G)\leq\lambda(G)$, and in particular when $G$ is an $s\times s$ square grid, then $\lambda(G)=s$. Chekuri and Chuzhoy (2016) show that there is a polynomial relationship between the treewidth and square grid minor size of a graph [5]. Since $\lambda(G)\geq s(G)$, their results demonstrate a polynomial relationship between the treewidth and the net order of a graph.

Smith [14] has a polynomial time algorithm to compute the minimum cover of a net, giving a lower bound for $BN(G)$. In Section 4, we present a faster such algorithm. Given a net of a planar graph, $G$, we construct an $O(n^{2})$ time algorithm, Net-Alg, whose output is a minimum cover. We characterize a cover as a tree drawn in the plane which may go through the faces, edges or vertices of $G$. Net-Alg proceeds by the shortest path algorithm from Henzinger, et al. [8] and inspiration from Dreyfus [6] to find a Steiner tree that meets all three sides of the net.

We use nets to replace square grids in the work of Bodlaender, Grigoriev and Koster [4]. They construct a rooted-search-tree algorithm to find lower bounds on the treewidth of a graph. In particular, their algorithm finds a square grid minor of size $LB_{2}$. The claim is that $LB_{2}\geq s(G)/4$; however as we will show in Section 6, the proof shows that $LB_{2}\geq s(G)/5$ only. Grigoriev [7] writes a new algorithm, using ideas from [4], to construct a tree decomposition of a graph and thus gets an upper bound on treewidth. We adapt ideas from both [4] and [7] in our rooted-search tree algorithm, BT-Alg. Our algorithm uses nets to achieve both lower and upper bounds on bramble number, as seen in the following theorem.

The outline of the paper is as follows. In Section 2 we do some preliminaries. In Section 3 we define nets and give an alternative characterization of a cover of net. In Section 4 we construct Net-Alg. In Section 5 we construct BT-Alg and prove Theorem 5.7. In Section 6 we correct the lower bound in Bodlaender et al. We make some concluding remarks in Section 7.

In this paper, every graph will be simple; for a graph $G$, we let $V(G)$ denote its vertex set and $E(G)$ denote its edge set. For each vertex, $u\in V(G)$, we let $N_{G}(u)$ denote its neighborhood, the set of vertices adjacent to $u$ in $G$. For any subset of vertices, $U\subseteq V(G)$, let $G[U]$ denote the subgraph of $G$ induced by $U$. If $G[U]$ is connected, then we say the vertex set, $U$, is connected in $G$. We denote the induced subgraph, $G[V(G)\backslash U]$, by $G-U$. We will refer to the power set of vertices in $G$ as $\mathcal{P}(V(G))$. If $T$ is a tree and $s,t\in V(T)$, then let $sTt$ denote the unique path from $s$ to $t$ in $T$. Similarly, if $W=(u_{0},...,u_{n})$ is a walk in a graph, then for any $0\leq i\leq j\leq n$, let $u_{i}Wu_{j}$ denote the subwalk from $u_{i}$ to $u_{j}$ in $W$. If $W$ is a closed walk, let $(u_{j}Wu_{n},u_{1}Wu_{i})$ denote the concatenated walk, $(u_{j},u_{j+1},...,u_{n},u_{1},u_{2},...,u_{i})$.

The width of a tree decomposition is defined as $\text{width}(T,\beta):=\max\{|\beta(t)|:t\in V(T)\}-1$ and the treewidth of a graph $G$ is

Using ideas from Reed in [9], we present an equivalent definition for a tree decomposition by looking at the inverse mapping:

The proof of lemma 2.2 follows directly from the definition of a tree decomposition. Now we will give the precise definition of a bramble.

For a simple example of a bramble in a graph, we can take the collection of vertices (as singleton sets) from a clique. The order of this bramble is the number of vertices in the clique. In particular, the bramble number of any graph is bounded below by its clique number. However, this example does not take advantage of the ability to have intersection among sets. Allowing sets in a bramble to intersect gives a much more robust class of obstructions to low treewidth.

When talking about graph embeddings, we follow the conventions of West [15]. In particular, we refer to a graph embedded in the plane as a plane graph, and for a plane graph, $G$, a face, $F$, of $G$ is a maximal region in the plane such that for any two points in $F$, there is a curve avoiding $G$ connecting those points. For computational reasons, we point out that in any plane graph, if $v$ is the number of vertices, $e$ is the number of edges and $f$ is the number of faces, then $e\leq 3v-6$ and $f\leq 2v-4$. Therefore, any computation that iterates on vertices, edges and faces of a graph still runs in $O(v)$-time. With this in mind, for any planar graph we let $|G|=|V(G)|$.

We are ready to define a natural family of three-sided brambles that occur in plane graphs, generalizing the bramble of crosses in a square grid graph described in the introduction.

A 3-frame decomposes $W$ into three subwalks with overlapping endpoints.

Throughout the rest of the paper we refer to the vertex sets of the sides of the 3-frame by colors:

That is, an $\mathcal{F}$-vine has at least one $blue$, one $red$ and one $yellow$ vertex.

The graph in Figure 1 is given a frame with sides $\{a,b\}$, $\{b,c\}$ and $\{c,d,a\}$. This frame defines a net whose elements are connected subsets of vertices, each of which intersects all three sides. The minimal elements of this net are $\{a,b\}$, $\{b,c\}$, $\{a,c,d\}$ and $\{b,d,e\}$. Figure 2 shows a 3-frame of a 6-triangular grid.

Note that the vertex sets $blue$, $red$ and $yellow$ are not disjoint, sharing at least one vertex between pairs. In the case where $W$ is a cycle, each pair of colors intersect on exactly one vertex, but both $W$ and our choice of $j$ and $k$ may cause more intersection between the sides.

The proof of the lemma demonstrates how the definition of a net takes advantage of the topology of the plane to guarantee intersection between any two vines. We will continue to use properties of our embedding to understand the order of these brambles.

In order to understand what a minimum size cover of a net looks like, we consider what would topologically prevent a vine touching all three sides of the 3-frame. As we saw in the proof of lemma 3.5, any two vines have non-trivial intersection. That is, each vine in a net is itself a cover of the bramble.

On the other hand, a sparse plane graph may not take full advantage of the space to find vines with few vertices. See Figure 3. For example, consider a cycle on 15 vertices, $C_{15}=(c_{0},c_{1},...,c_{15})$. We could evenly divide the cycle into a 3-frame: $(C_{15},5,10)$, and any $(C_{15},5,10)$-vine would use at least five vertices. However, a set of two vertices, $\{c_{5},c_{10}\}$, covers the $(C_{15},5,10)$-net. We can use the embedding to understand why this set covers the bramble by adding an edge between $c_{5}$ and $c_{10}$. This additional edge would make $\{c_{5},c_{10}\}$ a $(C_{15},5,10)$-vine, and any embedding of $C_{15}$ would afford us the space to make such an edge.

To account for these latent vines, we need to pay attention to what connections are possible through faces of the embedding. We now define a new plane graph obtained from a plane graph that inherits connectivity from the bounded faces in our embedding.

Since each edge of $G$ is contained in at most two faces of $G$, a greedy search on edges would find all faces in $O(|G|)$ time. Notice that $\mathcal{F}$ is also a 3-frame of $\widehat{G}$, so $\mathbf{N}(G,\mathcal{F})\subseteq\mathbf{N}({\widehat{G}},\mathcal{F})$. The face graph is something like a combination of a plane graph with its dual and will give us a useful property concerning the connectedness of the graph.

The following definition is inspired by a similar definition used by Robertson and Seymour in [11].

The Jordan Curve Theorem implies for any two paths in a connected plane graph, if the endpoints of one cross the endpoints of another in the peripheral walk, then the two paths share a common vertex. We use this fact in the following useful characterization of separating sets in face graphs.

We are ready to give an alternative characterization for a vertex set covering a net in a plane graph.

Using the topology of a planar embedding, the problem of finding a cover of a net is equivalent to finding a connected subgraph of the face graph that intersects all three sides of the 3-frame. In the next section, we algorithmically minimize such a cover using shortest paths.

Let $G$ be a connected plane graph with a 3-frame $\mathcal{F}$. In order to develop an algorithm that finds a minimum order cover of a net, define the indicator function $\mathbf{1}_{G}$ on $V(\widehat{G})$ such that

Let $G_{\leftrightarrow}$ denote a directed graph obtained from $G$ by replacing each edge with two directed edges of opposite orientation. Any directed path in $G_{\leftrightarrow}$ will then correspond to an undirected path of $G$ and vice versa. Moreover, two directed paths are vertex-disjoint in $G_{\leftrightarrow}$ just in case the corresponding paths in $G$ are vertex-disjoint.

From $\mathbf{1}_{G}$, we obtain a weight function, $\mathbf{1}_{G\leftrightarrow}$, on the edges of $\widehat{G}_{\leftrightarrow}$, where each edge weight is given by the weight of its terminal vertex under $\mathbf{1}_{G}$. We determine the weight of a directed path in $\widehat{G}_{\leftrightarrow}$ to be the sum of its edge weights. Then $\mathbf{1}_{G\leftrightarrow}$ gives us the following distance function on $\widehat{G}_{\leftrightarrow}$.

We use this distance function to define a specific structure possessed by any “minimum weight” $\mathcal{F}$-vine in $\widehat{G}$. This structure is essentially the three vertex case of the Steiner tree characterization given by Dreyfus and Wagner in [6].

With this structural characterization in hand, we can search for a minimum size cover of a net using the single-source shortest path algorithm for planar graphs by Henzinger et al. [8]

Now that we have an algorithm for finding a minimum net cover in a particular framed plane graph, we will use it to search a graph (and subgraphs) for large order nets.

We are interested not only in the order of a net in a particular framing of a plane graph, but more importantly in what nets are possible in subgraphs of that graph. Any net is hightly sensitive to the walk peripheral to the unbounded face of the embedding — if this walk is something simple like a 3-cycle, it severely limits the complexity of the net. However, higher order nets may be lurking in the interior of this embedding.

Lemma 5.2 implies that for any connected component $G^{\prime}$ of $G-(Y\cap V(G))$, a closed walk peripheral to $G^{\prime}$ can be decomposed into two internally disjoint subwalks overlapping on their endpoints. One of these subwalks is a subwalk of $W$ and the other has no interior vertices in $W$; that is, the interior vertices of this subwalk are uncolored in $G$.

We are ready to present an algorithm for finding a high order net in a subgraph of $G$. Our algorithm is inspired by Bodlaender, Grigoriev and Koster’s algorithm for finding a large square grid as a minor of a planar graph [4]. See Figure 5 for an example of its output.