Csaba Biró

The indeque number of a graph is largest set of vertices that induce an independent set of cliques. We study the extremal value of this parameter for the class and subclasses of planar graphs, most notably for forests and graphs of pathwidth at most $2$.

The interval count problem, a classical question in the study of interval orders, was introduced by Ronald Graham in the 1980s. This problem asks: given an interval order $P$, what is the minimum number of distinct interval lengths required to construct an interval representation of $P$? Interval orders that can be represented with just one interval length are known as semiorders, and their characterizations are well known. However, the characterization of interval orders that require at most $k$ interval lengths -- termed $k$-count interval orders -- remains an open and challenging problem for $k\geq 2$. Our investigation into $2$-count interval orders led us naturally to consider a related problem, interval representations of permutations, which we introduce in this paper. Specifically, we characterize permutations that have a $2$-count interval representation. We prove that a permutation admits a $2$-count interval representation if and only if its longest decreasing subsequences have length at most $2$. For larger values of $k$, however, a similar characterization does not hold. There are permutations that do not permit a $3$-count interval representation despite having decreasing subsequences of length at most $3$. Characterizing $k$-count permutations remains open for $k \geq 3$. The $k$-count permutation representation problem appears to capture essential aspects of the broader problem of characterizing $k$-count interval orders. To support this connection, we apply our findings on interval representations of permutations to demonstrate that a height-$3$ interval order is $2$-count if and only if it has depth at most $2$, where the depth of an interval order refers to the length of the longest nested chain of intervals required in any interval representation of the order.

The length polyhedron $Q_P$ of an interval order $P$ is the convex hull of integral vectors representing the interval lengths in interval representations of $P$. This polyhedron has been studied by various authors, including Fishburn and Isaak. Notably, $Q_P$ forms a pointed affine cone, a property inherited from being a projection of the representation polyhedron, a structure explored also by Doignon and Pauwels. The apex of the length polyhedron corresponds to Greenough's minimal endpoint representation, which is, in fact, the length vector of the canonical interval representation -- an interval representation that minimizes the sum of the interval lengths. Building on a combinatorial perspective of canonical representations, we refine Isaak's graph-theoretical model by introducing a new and simpler directed graph. From directed cycles of this key digraph, we extract a linear system of inequalities that precisely characterizes the length polyhedron $Q_P$. This combinatorial approach also reveals the unique Hilbert basis of the polyhedron. We prove that the intersection graph of the sets corresponding to these binary rays are Berge graphs; therefore they are perfect graphs. As a result, for interval orders with bounded width, the length polyhedron has a polynomial-sized Hilbert basis, which can be computed in polynomial time. We also provide an example of interval orders with a Hilbert basis of exponential size. In a companion paper we determine the Schrijver system for the length polyhedron. We conclude with open problems.