Vertex Deletion into Bipartite Permutation Graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines ?? and ??, 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 NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980]. 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 f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Automorphisms and endomorphisms of first-order structures

In this thesis, we consider questions relating to automorphisms and endomorphisms of countable, relational first-order structures M, with a particular emphasis on bimorphism monoids. We determine semigroup-theoretic results for three types of endomorphism monoid onM, along with generation results whenMis the random graph R or the discrete linear order (N;_). In addition, we introduce three types of partial map monoid ofM, and prove some semigroup-theoretic and generation results in these cases. We introduce the idea of a permutation monoid, and characterise the closed submonoids of the infinite symmetric group Sym(N). Following this, we turn our attention the idea of oligomorphic transformation monoids, and expand on the existing results by considering a range of notions of homomorphismhomogeneity as introduced by Lockett and Truss in 2012. Furthermore, we show that for any finite group G, there exists an oligomorphic permutation monoid with group of units isomorphic to G. The main result of the thesis is an analogue of Fra¨ıss´e’s theorem covering twelve of the eighteen notions of homomorphism-homogeneity; this contains both Fra¨ıss´e’s theorem, and a version of this for MM-homogeneous structures by Cameron and Neˇsetˇril in 2006, as corollaries. This is then used to determine the extent to which some well-known countable homogeneous structures are also homomorphism-homogeneous. Finally, we turn our attention to MB-homogeneous graphs and digraphs. We begin by classifying those homogeneous graphs that are also MB-homogeneous. We then determine an example of an MB-homogeneous graph not in this classification, and use the idea behind this construction to demonstrate 2@0 many non-isomorphic examples of MB-homogeneous graphs. We also give 2@0 many non-isomorphic examples of MB-homogeneous digraphs