Graph minors. V. Excluding a planar graph

Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

“…tree-width or branch-width) and arbitrarily large maximum deficiency. In particular, the maximum deficiency of formulas whose incidence graphs are grids is at most 1, but the tree-width of n × n grids is n. The significance of this discrepancy is further emphasized by Robertson and Seymour's deep Excluded Grid Theorem [25], which states that graphs of high tree-width necessarily have large square grids as minors. (2) Maximum deficiency can be computed in polynomial time by matching algorithms [12].…”

Graph Operations Characterizing Rank-Width and Balanced Graph Expressions

“…Here twd, cwd and mcwd denote respectively tree-width [17], clique-width [6] and m-clique-width (we recall below the definition of m-clique-width [7]). The following constants will be used: for A ⊆ L we let A be a constant denoting the graph G with single vertex x and δ G (x) = A.…”

“…Graph complexity measures like tree-width [17], clique-width [6], NLC-width [18] and rank-width [16] are important parameters for the construction of polynomial algorithms. Every graph property expressible by a formula of MS (Monadic SecondOrder) logic has a Fixed Parameter Linear algorithm if tree-width is taken as parameter and a Fixed Parameter Cubic algorithm if clique-width (equivalently rankwidth) is taken as parameter.…”

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

“…Forbidden minors are minors which a class of graphs is known to exclude [24]: Wagner's theorem established that planar graphs cannot contain K 5 or K 3,3 minors [25], while the works of Robertson and Seymour [26][27][28][29][30] develop the theory of forbidden minors for planar and non-planar graphs (see also the review in Ref. [31]).…”