We unify several seemingly different graph and digraph classes under one umbrella. These classes are all broadly speaking different generalizations of interval graphs, and include, in addition to interval graphs, also adjusted interval digraphs, thre
shold graphs, complements of threshold tolerance graphs (known as `co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing loops. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above example graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having all loops, or having a symmetric edge-set, or being bipartite. (For instance co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric.) We also offer some discussion of recognition algorithms and characterizations, saving the details for future papers.
A semiorder is a model of preference relations where each element $x$ is associated with a utility value $alpha(x)$, and there is a threshold $t$ such that $y$ is preferred to $x$ iff $alpha(y) > alpha(x)+t$. These are motivated by the notion that th
ere is some uncertainty in the utility values we assign an object or that a subject may be unable to distinguish a preference between objects whose values are close. However, they fail to model the well-known phenomenon that preferences are not always transitive. Also, if we are uncertain of the utility values, it is not logical that preference is determined absolutely by a comparison of them with an exact threshold. We propose a new model in which there are two thresholds, $t_1$ and $t_2$; if the difference $alpha(y) - alpha(x)$ less than $t_1$, then $y$ is not preferred to $x$; if the difference is greater than $t_2$ then $y$ is preferred to $x$; if it is between $t_1$ and $t_2$, then then $y$ may or may not be preferred to $x$. We call such a relation a double-threshold semiorder, and the corresponding directed graph $G = (V,E)$ a double threshold digraph. Every directed acyclic graph is a double threshold graph; bounds on $t_2/t_1$ give a nested hierarchy of subclasses of the directed acyclic graphs. In this paper we characterize the subclasses in terms of forbidden subgraphs, and give algorithms for finding an assignment of of utility values that explains the relation in terms of a given $(t_1,t_2)$ or else produces a forbidden subgraph, and finding the minimum value $lambda$ of $t_2/t_1$ that is satisfiable for a given directed acyclic graph. We show that $lambda$ gives a measure of the complexity of a directed acyclic graph with respect to several optimization problems that are NP-hard on arbitrary directed acyclic graphs.
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatr
ices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any binary matrix that does not have the consecutive-ones property.
A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph $widetilde{G}$ is a digraph obtained from a sequence of vertex complement operations on $G
$. Dahlhaus et al. showed that, given an adjacency-list representation of $widetilde{G}$, depth-first search (DFS) on $G$ can be performed in $O(n + widetilde{m})$ time, where $n$ is the number of vertices and $widetilde{m}$ is the number of edges in $widetilde{G}$. To achieve this bound, their algorithm makes use of a somewhat complicated stack-like data structure to simulate the recursion stack, instead of implementing it directly as a recursive algorithm. We give a recursive $O(n+widetilde{m})$ algorithm that uses no complicated data-structures.
The interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each intersecting pair of intervals. A probe interval graph is a variant that is motivated by an application to genomics, where the interval
s are partitioned into two sets: probes and non-probes. The graph has an edge between two vertices if they intersect and at least one of them is a probe. We give a linear-time algorithm for determining whether a given graph and partition of vertices into probes and non-probes is a probe interval graph. If it is, we give a layout of intervals that proves this. We can also determine whether the layout of the intervals is uniquely constrained within the same time bound. As part of the algorithm, we solve the consecutive-ones probe matrix problem in linear time, develop algorithms for operating on PQ trees, and give results that relate PQ trees for different submatrices of a consecutive-ones matrix.
We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Gamm
a-circular-arc graphs, proper circular-arc graphs and convex-round graphs.
