Subgraph densities and scaling limits of random graphs with a prescribed modular decomposition

We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a consequence, we obtain the convergence of a uniform random graph in such classes to a Brownian limit object in the space of graphons. Our proofs rely on combinatorial arguments, computing generating series using the symbolic method and deriving asymptotics using singularity analysis.Comment: 32 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:2301.1360

Representation of graphs by OBDDs

AbstractRecently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P4s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements

Reconstructing Gene Trees From Fitch's Xenology Relation

Two genes are xenologs in the sense of Fitch if they are separated by at least one horizontal gene transfer event. Horizonal gene transfer is asymmetric in the sense that the transferred copy is distinguished from the one that remains within the ancestral lineage. Hence xenology is more precisely thought of as a non-symmetric relation: $y$ is xenologous to $x$ if $y$ has been horizontally transferred at least once since it diverged from the least common ancestor of $x$ and $y$. We show that xenology relations are characterized by a small set of forbidden induced subgraphs on three vertices. Furthermore, each xenology relation can be derived from a unique least-resolved edge-labeled phylogenetic tree. We provide a linear-time algorithm for the recognition of xenology relations and for the construction of its least-resolved edge-labeled phylogenetic tree. The fact that being a xenology relation is a heritable graph property, finally has far-reaching consequences on approximation problems associated with xenology relations

Finding combinatorial structures

In this thesis we answer questions in two related areas of combinatorics: Ramsey theory and asymptotic enumeration. In Ramsey theory we introduce a new method for finding desired structures. We find a new upper bound on the Ramsey number of a path against a kth power of a path. Using our new method and this result we obtain a new upper bound on the Ramsey number of the kth power of a long cycle. As a corollary we show that, while graphs on n vertices with maximum degree k may in general have Ramsey numbers as large as ckn, if the stronger restriction that the bandwidth should be at most k is given, then the Ramsey numbers are bounded by the much smaller value. We go on to attack an old conjecture of Lehel: by using our new method we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new method replaces their use of the Regularity Lemma, and allows us to prove that for any n > 218000, whenever the edges of the complete graph on n vertices are two-coloured there exist disjoint monochromatic cycles covering all n vertices. In asymptotic enumeration we examine first the class of bipartite graphs with some forbidden induced subgraph H. We obtain some results for every H, with special focus on the cases where the growth speed of the class is factorial, and make some comments on a connection to clique-width. We then move on to a detailed discussion of 2-SAT functions. We find the correct asymptotic formula for the number of 2-SAT functions on n variables (an improvement on a result of Bollob´as, Brightwell and Leader [13], who found the dominant term in the exponent), the first error term for this formula, and some bounds on smaller error terms. Finally we obtain various expected values in the uniform model of random 2-SAT functions

Exploring Subexponential Parameterized Complexity of Completion Problems

Let ${\cal F}$ be a family of graphs. In the ${\cal F}$-Completion problem, we are given a graph $G$ and an integer $k$ as input, and asked whether at most $k$ edges can be added to $G$ so that the resulting graph does not contain a graph from ${\cal F}$ as an induced subgraph. It appeared recently that special cases of ${\cal F}$-Completion, the problem of completing into a chordal graph known as Minimum Fill-in, corresponding to the case of ${\cal F}=\{C_4,C_5,C_6,\ldots\}$, and the problem of completing into a split graph, i.e., the case of ${\cal F}=\{C_4, 2K_2, C_5\}$, are solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$. The exploration of this phenomenon is the main motivation for our research on ${\cal F}$-Completion. In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that: - The problem Trivially Perfect Completion is solvable in parameterized subexponential time $2^{O(\sqrt{k}\log{k})}n^{O(1)}$, that is ${\cal F}$-Completion for ${\cal F} =\{C_4, P_4\}$, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where ${\cal F} = \{2K_2, C_4\}$, and Threshold Completion, where ${\cal F} = \{2K_2, P_4, C_4\}$, are also solvable in time $2^{O(\sqrt{k}\log{k})} n^{O(1)}$. We complement our algorithms for ${\cal F}$-Completion with the following lower bounds: - For ${\cal F} = \{2K_2\}$, ${\cal F} = \{C_4\}$, ${\cal F} = \{P_4\}$, and ${\cal F} = \{2K_2, P_4\}$, ${\cal F}$-Completion cannot be solved in time $2^{o(k)} n^{O(1)}$ unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of ${\cal F}$-Completion problems for ${\cal F}\subseteq\{2K_2, C_4, P_4\}$.Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in the proceedings of STACS'1