In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F , the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F . Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when p...

We call an edge e of a perfect graph G critical if G e is imperfect and call e anticritical if G + e is imperfect. The present paper surveys several questions in this context. We ask in which perfect graphs critical and anticritical edges occur and how to detect such edges. The main result by Hougardy & Wagler [32] shows that a graph does not admit any critical edge if and only if it is Meyniel...

it is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph increases exactly by one, as it happens to its induced matching number and (castelnuovo-mumford) regularity. as a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph g are equal provided g is well-covered bipa...

Sidorenko’s conjecture states that the number of copies a bipartite graph H in G is asymptotically minimised when quasirandom graph. A notorious example where this remains open = K5,5C10. It was even unknown whether possesses strictly stronger, weakly norming property.We take step towards understanding K5,5C10 by proving it not norming. More generally, we show ‘twisted’ blow-ups cycles, which i...

In this paper we consider module-composed graphs, i.e. graphs which can be defined by a sequence of single vertex insertions v1, . . . , vn, such that the neighbourhood of vertex vi, 2 ≤ i ≤ n, forms a module (a homogenous set) of the graph defined by vertices v1, . . . , vi−1. Module composed graphs are HHDS-free and thus homogeneous orderable, weakly chordal, and perfect. Every bipartite dist...

A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least 5 or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by C...

altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. we now provide a generalization of the altan concept, applicable to any graph. several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. among these ar...

Hoang, C.T., F. Maffray, S. Olariu and M. Preissmann, A charming class of perfectly orderable graphs, Discrete Mathematics 102 (1992) 67-74. We investigate the following conjecture of VaSek Chvatal: any weakly triangulated graph containing no induced path on five vertices is perfectly orderable. In the process we define a new polynomially recognizable class of perfectly orderable graphs called ...