Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

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

Graphs with at most two moplexes

A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: Unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval. In this work we initiate an investigation of $k$-moplex graphs, which are defined as graphs containing at most $k$ moplexes. Of particular interest is the smallest nontrivial case $k=2$, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected $2$-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected $2$-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets

Intersections and Representations of Graphs

Given two graphs G and H sharing the same vertex set, the edge-intersection spectrum of G and H is the set of possible sizes of the intersection of the edge sets of both graphs. For example, the spectrum of two copies of the cycle C5 is {0, 2, 3, 5}, and the spectrum of two copies of the star K1,r is {1, r}. The intersection spectrum was initially studied for designs by Lindner and Fu and others and was originally extended to graphs by Eric Mendelsohn. Several examples are studied, both when G and H are isomorphic and when they are not isomorphic. It will also be shown any set S of positive integers is the edge-intersection spectrum of some pair of connected graphs. The other two chapters cover the area of conflict-tolerance graph representations. These representations consist of rules for measuring the rank and tolerance of each vertex, and for determining if two vertices are in conflict, by combining and comparing the ranks and tolerances of the vertices. The edge set of the graph is then the pairs of vertices which are in conflict. In the odd-intersection interval model, each vertex is represented by a subpath of a host path P, and two vertices are in conflict if and only if their corresponding subpaths intersect in an odd number of nodes. This model is not universal; in particular, the complete 4-partite graph K3,3,3,3 is a minimal forbidden subgraph. The parity of the subpaths affects the representation; graphs in which the subpaths are of a fixed odd order are strongly chordal (in fact, these are precisely the unit interval graphs), and graphs in which the subpaths are all of even order are bipartite. The converse of the latter statement is not true, as it will be shown that the number of bipartite graphs is asymptotically larger than the number of possible representations. A cross-comparison graph model\u3c\italic\u3e is one in which each vertex v is assigned a rank rv and a tolerance tv, and two vertices u and v are in conflict if rv ≥ tu and ru ≥ tv. Jamison showed that the cross-comparison model is universal, using n-dimensional vectors and coordinatewise comparison to represent a graph on n vertices. The inefficiency of a vector representation is the smallest number of dimensions d for which we can represent a graph G using d-dimensional vectors. The set of all graphs which can be represented using one-dimensional vectors (efficient cross-comparison graphs) is precisely the set of graphs which are the complement of a threshold tolerance graph (known as co-TT graphs; these were defined by Monma, Reed, and Trotter in 1988). The efficient cross-comparison graphs are characterized as those graphs which are chordal, and contain no strongly asteroidal triple --- a set of three vertices, such that there is a path between any two of these vertices which contains no neighbor of the third, and which does not contain two consecutive vertices adjacent to every neighbor of the third. In addition, a graph with a d-dimensional vector representation is the intersection of d efficient graphs. The inefficiency of G is bounded below by its chordality, and above by its boxicity; both of these bounds are tight. In addition, the graph Kn(2) has chordality and boxicity (and thus inefficiency) equal to n, and this is known to be the upper bound for boxicity, which shows that the efficiency of a graph is at most half its order, and that this bound is tight

Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes