Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

On a unique tree representation for P4-extendible graphs

AbstractSeveral practical applications in computer science and computational linguistics suggest the study of graphs that are unlikely to have more than a few induced paths of length three. These applications have motivated the notion of a cograph, defined by the very strong restriction that no vertex may belong to an induced path of length three. The class of P4-extendible graphs that we introduce in this paper relaxes this restriction, and in fact properly contains the class of cographs, while still featuring the remarkable property of admitting a unique tree representation. Just as in the case of cographs, the class of P4-extendible graphs finds applications to clustering, scheduling, and memory management in a computer system. We give several characterizations for P4-extendible graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P4-extendible graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time

A tree representation for P4-sparse graphs

AbstractA graph G is P4-sparse if no set of five vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize both the class of cographs and the class of P4-reducible graphs. We give several characterizations for P4-sparse graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P4-sparse graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time

Maximal independent sets and maximal matchings in series-parallel and related graph classes

The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and seriesparallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].Postprint (author's final draft

Coloring polygon visibility graphs and their generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo- polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3 · 4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo- visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a coloring with the claimed number of colors can be computed in polynomial time

Obstructions to Faster Diameter Computation: Asteroidal Sets

Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than $\alpha$ pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every $m$-edge graph in $Ext_{\alpha}$ can be computed in deterministic ${\cal O}(\alpha^3 m^{3/2})$ time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive $+1$-approximation of all vertex eccentricities in deterministic ${\cal O}(\alpha^2 m)$ time. This is in sharp contrast with general $m$-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in ${\cal O}(m^{2-\epsilon})$ time for any $\epsilon > 0$. As important special cases of our main result, we derive an ${\cal O}(m^{3/2})$-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an ${\cal O}(k^3m^{3/2})$-time algorithm for this problem on graphs of asteroidal number at most $k$. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions