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In this paper we investigate some problems related to the Helly properties of circular-arc graphs, which are defined as intersection graphs of arcs of a fixed circle. As such, circular-arc graphs are among the simplest classes of intersection graphs whose models might not satisfy the Helly property. In particular, some cliques of a circular-arc graph might be Helly in some but not all arc intersection models of the graph. Our first result is an alternative proof of a theorem by Lin and Szwarcfiter which asserts that for every circular-arc graph $G$ either every normalized model of $G$ satisfies the Helly property or no normalized model of $G$ satisfies this property. Further, we study the Helly properties of a single clique of a circular-arc graph $G$. We divide the cliques of $G$ into three types: a clique $C$ of $G$ is always-Helly/always-non-Helly/ambiguous if $C$ is Helly in every/no/(some but not all) normalized model of $G$. We provide a combinatorial description for the cliques of each type, and based on it, we devise a polynomial time algorithm which determines the type of a given clique. Finally, we study the Helly Cliques problem, in which we are given an $n$-vertex circular-arc graph $G$ and some of its cliques $C_1, \ldots, C_k$ and we ask if there is an arc intersection model of $G$ in which all the cliques $C_1, \ldots, C_k$ satisfy the Helly property. We show that: (1) the Helly Cliques problem admits a $2^{O(k\log{k})}n^{O(1)}$-time algorithm (that is, it is FPT when parametrized by the number of cliques given in the input), (2) assuming Exponential Time Hypothesis (ETH), the Helly Cliques problem cannot be solved in time $2^{o(k)}n^{O(1)}$, (3) the Helly Cliques problem admits a polynomial kernel of size $O(k^6)$. All our results use a data structure, called a PQM-tree, which maintains all normalized models of a circular-arc graph $G$.

The most elusive problem around the class of circular-arc graphs is identifying all minimal graphs that are not in this class. The main obstacle is the lack of a systematic way of enumerating these minimal graphs. McConnell [FOCS 2001] presented a transformation from circular-arc graphs to interval graphs with certain patterns of representations. We fully characterize these interval patterns for circular-arc graphs that are split graphs, thereby building a connection between minimal split graphs that are not circular-arc graphs and minimal non-interval graphs. This connection enables us to identify all minimal split graphs that are not circular-arc graphs. As a byproduct, we develop a linear-time certifying recognition algorithm for circular-arc graphs when the input is a split graph.

In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph $H$, the class of $H$-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of $H$. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of $H$-graphs for different graphs $H$. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and Zeman showed, for every fixed tree $T$, a polynomial-time algorithm recognizing $T$-graphs. Tucker showed a polynomial time algorithm recognizing $K_3$-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed that recognition of $H$-graphs is $NP$-hard if $H$ contains two different cycles sharing an edge. The main two results of this work narrow the gap between the $NP$-hard and $P$ cases of $H$-graphs recognition. First, we show that recognition of $H$-graphs is $NP$-hard when $H$ contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing $L$-graphs, where $L$ is a graph containing a cycle and an edge attached to it ($L$-graphs are called lollipop graphs). Our work leaves open the recognition problems of $M$-graphs for every unicyclic graph $M$ different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows. Firstly, the recognition of $M$-graphs, where $M$ is a fixed unicyclic graph, admits a polynomial time algorithm if we restrict the input to graphs containing particular holes (hence recognition of $M$-graphs is probably most difficult for chordal graphs). Secondly, the recognition of medusa graphs, which are defined as the union of $M$-graphs, where $M$ runs over all unicyclic graphs, is $NP$-complete.

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $\ell_1$ and $\ell_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In the the bipartite permutation vertex deletion problem we ask 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 but it does admit an FPT algorithm parameterized by $k$. In this paper we study the kernelization of this problem and show that it admits a polynomial kernel with $O(k^{62})$ vertices.

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, 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 NP-complete by the classical result of Lewis and Yannakakis. 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 $O(9^k\cdot n^9)$, and also give a polynomial-time 9-approximation algorithm., Comment: Extended abstract accepted to International Symposium on Parameterized and Exact Computation (IPEC'20)