Obstructions for linear rankwidth at most 1
We provide a characterization of graphs of linear rankwidth at most 1 by minimal excluded vertex-minors.
cs.DM↗
arXiv subjects
Isolde Adler
Publications and source records attributed to Isolde Adler.
Nowhere dense graph classes, stability, and the independence property
A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of cliques that occur as (topological) r-minors. We observe that this tameness notion from algorithmic graph theory is essentially the earlier stability theoretic notion of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence property.
math.LO↗
A lower bound for the tree-width of planar graphs with vital linkages
The disjoint paths problem asks, given an graph G and k + 1 pairs of terminals (s_0,t_0), ...,(s_k,t_k), whether there are k+1 pairwise disjoint paths P_0, ...,P_k, such that P_i connects s_i to t_i. Robertson and Seymour have proven that the problem can be solved in polynomial time if k is fixed. Nevertheless, the constants involved are huge, and the algorithm is far from implementable. The algorithm uses a bound on the tree-width of graphs with vital linkages, and deletion of irrelevant vertices. We give single exponential lower bounds both for the tree-width of planar graphs with vital linkages, and for the size of the grid necessary for finding irrelevant vertices.
cs.DS↗
Games for width parameters and monotonicity
We introduce a search game for two players played on a "scenario" consisting of a ground set together with a collection of feasible partitions. This general setting allows us to obtain new characterisations of many width parameters such as rank-width and carving-width of graphs, matroid tree-width and GF(4)-rank-width. We show that the monotone game variant corresponds to a tree decomposition of the ground set along feasible partitions. Our framework also captures many other decompositions into "simple" subsets of the ground set, such as decompositions into planar subgraphs. Within our general framework, we take a step towards characterising monotone search games. We exhibit a large class of "monotone" scenarios, i.e. of scenarios where the game and its monotone variant coincide. As a consequence, determining the winner is in NP for these games. This result implies monotonicity for all our search games, that are equivalent to branch-width of a submodular function. Finally, we include a proof showing that the matroid tree-width of a graphic matroid is not larger than the tree-width of the corresponding graph. This proof is considerably shorter than the original proof and it is purely graph theoretic.
cs.DM↗
A note on clique-width and tree-width for structures
We give a simple proof that the straightforward generalisation of clique-width to arbitrary structures can be unbounded on structures of bounded tree-width. This can be corrected by allowing fusion of elements.
cs.LO↗