A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $$\ell _1$$
ℓ
1
and $$\ell _2$$
ℓ
2
, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is $$\mathsf {NP}$$
NP
-complete by the classical result of Lewis and Yannakakis [20]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time $${\mathcal {O}}(9^k \cdot n^9)$$
O
(
9
k
·
n
9
)
, and also give a polynomial-time 9-approximation algorithm.
Tuza famously conjectured in 1981 that in a graph without k + 1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colourable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.
We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. For most of the graphs we prove the exact value, while for all the remaining ones we provide very close lower and upper bounds. It occurs that, for some graphs, the structure of extremal constructions maximizing density of its induced copies is very sophisticated and complex.
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