On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer $k$, $B_k$-EPG graphs are defined as EPG graphs admitting a model in which each path has at most $k$ bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a $B_4$-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is $B_3$-EPG, and that there exist circular-arc graphs which are not $B_2$-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define $B_k$-EPR graphs, $k\geq 0$, the same way as $B_k$-EPG graphs. Circular-arc graphs are clearly $B_4$-EPR graphs and we will show that there exist circular-arc graphs that are not $B_3$-EPR graphs. We also show that normal circular-arc graphs are $B_2$-EPR graphs and that there exist normal circular-arc graphs that are not $B_1$-EPR graphs. Finally, we characterize $B_1$-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs

Characterising and recognising game-perfect graphs

Consider a vertex colouring game played on a simple graph with $k$ permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game $g_B$, the breaker makes the first move. Our main focus is on the class of $g_B$-perfect graphs: graphs such that for every induced subgraph $H$, the game $g_B$ played on $H$ admits a winning strategy for the maker with only $\omega(H)$ colours, where $\omega(H)$ denotes the clique number of $H$. Complementing analogous results for other variations of the game, we characterise $g_B$-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise $g_B$-perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the International Colloquium on Graph Theory (ICGT) 201

Faster Algorithms for Computing Maximal 2-Connected Subgraphs in Sparse Directed Graphs

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with $\widetilde{O}(mn)$ time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced $O(n^2)$ time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time $O(m^{3/2})$, which further improves the running times for sparse graphs. The notion of 2-connectivity naturally generalizes to k-connectivity for $k>2$. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time $O(m^{3/2} \log{n})$, improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in $O(n^2 \log n)$ time.Comment: Revised version of SODA 2017 paper including details for k-edge-connected subgraph

Unit Interval Editing is Fixed-Parameter Tractable

Given a graph~$G$ and integers $k_1$, $k_2$, and~$k_3$, the unit interval editing problem asks whether $G$ can be transformed into a unit interval graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. We give an algorithm solving this problem in time $2^{O(k\log k)}\cdot (n+m)$, where $k := k_1 + k_2 + k_3$, and $n, m$ denote respectively the numbers of vertices and edges of $G$. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time $O(4^k \cdot (n + m))$. Another result is an $O(6^k \cdot (n + m))$-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time $O(6^k \cdot n^6)$.Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe

Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices

On rigidity, orientability and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G \in \mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q \leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.Comment: 32 pages, 1 figur

Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces

The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS '10), with complexity $2^{O(k)} O(n)$. We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. At the same time, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (SODA '95 / JGAA '99)