Distance-hereditary graphs

Obstructions for linear rank-width at most 1

“…, where rank is the rank function over GF (2). A rank-decomposition of a graph G is a pair (T, λ), where T is a cubic tree and λ : L(T ) → V (G) is a bijection.…”

“…A distance-hereditary graph is a graph G for which the distances between vertices in any connected, induced subgraph of G are the same as in G [11]. Every distance-hereditary graph can be constructed from a single vertex by a sequence of the following three dh-operations [2]: We denote a dh-operation acting on a vertex u of a graph G by ⊕ x (G, u), where x ∈ {p, t, f }.…”

“…Obstruction set characterizations are known for the class of graphs of rank-width at most 1 (i.e. for the distance hereditary graphs) [2] and for the class of circle graphs [4]. To obtain our results, we depend upon the known equivalence between graphs of linear rank-width at most 1 and thread graphs [9] and the fact that we can reduce the search for unknown obstructions to distance-hereditary graphs.…”

A Faster Implementation of a Parallel Tree Contraction Scheme and Its Application on Distance-Hereditary Graphs

“…1a. 2. RAKE: An operation removes all the leaves from T. An example of a single RAKE operation is shown in Fig.…”

Characterization and recognition of some opposition and coalition graph classes

“…Chvátal proved that < is a perfect order if and only if there is no chordless path with four vertices a, b, c, d and three edges ab, bc, cd (written P 4 abcd) with a < b and d < c. He also proved that perfectly orderable graphs are perfect. 1 The class of perfectly orderable graphs properly contains many important, well-known classes of perfect graphs such as chordal graphs and comparability graphs. Perfectly orderable graphs have been extensively studied in the literature; see Hoàng's comprehensive survey [11] for more information.…”