Path Decomposition - Bounds

Bounds

Every n-vertex graph with pathwidth k has at most k(n − k + (k − 1)/2)) edges, and the maximal pathwidth-k graphs (graphs to which no more edges can be added without increasing the pathwidth) have exactly this many edges. A maximal pathwidth-k graph must be either a k-path or a k-caterpillar, two special kinds of k-tree. A k-tree is a chordal graph with exactly n − k maximal cliques, each containing k + 1 vertices; in a k-tree that is not itself a (k + 1)-clique, each maximal clique either separates the graph into two or more components, or it contains a single leaf vertex, a vertex that belongs to only a single maximal clique. A k-path is a k-tree with at most two leaves, and a k-caterpillar is a k-tree in which the non-leaf vertices induce a k-path. In particular the maximal graphs of pathwidth one are exactly the caterpillar trees.

Since path-decompositions are a special case of tree-decompositions, the pathwidth of any graph is greater than or equal to its treewidth. The pathwidth is also less than or equal to the cutwidth, the minimum number of edges that crosses any cut between lower-numbered and higher-numbered vertices in an optimal linear arrangement of the vertices of a graph; this follows because the vertex separation number, the number of lower-numbered vertices with higher-numbered neighbors, can at most equal the number of cut edges. For similar reasons, the cutwidth is at most the pathwidth times the maximum degree of the vertices in a given graph.

Any n-vertex forest has pathwidth O(log n). For, in a forest, one can always find a constant number of vertices the removal of which leaves a forest that can be partitioned into two smaller subforests with at most 2n/3 vertices each. A linear arrangement formed by recursively partitioning each of these two subforests, placing the separating vertices between them, has logarithmic vertex searching number. The same technique, applied to a tree-decomposition of a graph, shows that, if the treewidth of an n-vertex graph G is t, then the pathwidth of G is O(t log n). Since outerplanar graphs, series-parallel graphs, and Halin graphs all have bounded treewidth, they all also have at most logarithmic pathwidth.

As well as its relations to treewidth, pathwidth is also related to clique-width and cutwidth, via line graphs; the line graph L(G) of a graph G has a vertex for each edge of G and two vertices in L(G) are adjacent when the corresponding two edges of G share an endpoint. Any family of graphs has bounded pathwidth if and only if its line graphs have bounded linear clique-width, where linear clique-width replaces the disjoint union operation from clique-width with the operation of adjoining a single new vertex. If a connected graph with three or more vertices has maximum degree three, then its cutwidth equals the vertex separation number of its line graph.

In any planar graph, the pathwidth is at most proportional to the square root of the number of vertices. One way to find a path-decomposition with this width is (similarly to the logarithmic-width path-decomposition of forests described above) to use the planar separator theorem to find a set of O(√n) vertices the removal of which separates the graph into two subgraphs of at most 2n/3 vertices each, and concatenate recursively-constructed path decompositions for each of these two subgraphs. The same technique applies to any class of graphs for which a similar separator theorem holds. Since, like planar graphs, the graphs in any fixed minor-closed graph family have separators of size O(√n), it follows that the pathwidth of the graphs in any fixed minor-closed family is again O(√n). For some classes of planar graphs, the pathwidth of the graph and the pathwidth of its dual graph must be within a constant factor of each other: bounds of this form are known for biconnected outerplanar graphs and for polyhedral graphs. For 2-connected planar graphs, the pathwidth of the dual graph less than the pathwidth of the line graph. It remains open whether the pathwidth of a planar graph and its dual are always within a constant factor of each other in the remaining cases.

In some classes of graphs, it has been proven that the pathwidth and treewidth are always equal to each other: this is true for cographs, permutation graphs, the complements of comparability graphs, and the comparability graphs of interval orders.

In any cubic graph, or more generally any graph with maximum vertex degree three, the pathwidth is at most n/6 + o(n), where n is the number of vertices in the graph. There exist cubic graphs with pathwidth 0.082n, but it is not known how to reduce this gap between this lower bound and the n/6 upper bound.