Combinatorics and Algorithms for Quasi-chain Graphs

Abstract

The class of quasi-chain graphs is an extension of the well-studied class of chain graphs. This latter class enjoys many nice and important properties, such as bounded clique-width, implicit representation, well-quasi-ordering by induced subgraphs, etc. The class of quasi-chain graphs is substantially more complex. In particular, this class is not well-quasi-ordered by induced subgraphs, and the clique-width is not bounded in it. In the present paper, we show that the universe of quasi-chain graphs is at least as complex as the universe of permutations by establishing a bijection between the class of all permutations and a subclass of quasi-chain graphs. This implies, in particular, that the induced subgraph isomorphism problem is NP-complete for quasi-chain graphs. On the other hand, we propose a decomposition theorem for quasi-chain graphs that implies an implicit representation for graphs in this class and efficient solutions for some algorithmic problems that are generally intractable.

Understanding lettericity I: a structural hierarchy

“…The cells are labelled Z ij , where the first index counts from left to right, and the second from bottom to top. 4 Definition 5. Let π be a permutation and M = (α ij ) an s × t matrix with entries in {0, ±1}.…”

“…It is interesting to note that the same very similar examples kept recurring. For instance, in [4], we identify minimal classes of unbounded lettericity among one-sided 2P 3 -free bipartite graphs. Our list consists of induced matchings, their bipartite complements, and a third class that we dubbed "double-chain graphs": the universal graphs for it look like prime chain graphs, except that each vertical edge is inflated to a 2K 2 .…”

“…The classes we are interested in are the class C k of all graphs C k,l and their induced subgraphs, and the class C k of their CC-complements. We note that the minimal classes identified in both [4] and [5] are, up to some complementations, exactly the cycles-in-a-chain construction C 4,l . Our first result is that C k and C k have unbounded lettericity (that is, they are our classes of bounded λ but unbounded lettericity).…”

“…Our first result is that C k and C k have unbounded lettericity (that is, they are our classes of bounded λ but unbounded lettericity). In view of [4] and [5], the proof of (a) Bags in a cycle with chain graphs between them.…”

“…Lo and behold! It is, up to complementation, the same 4-chain circuit from [4] and [5]. The natural follow-up question is: are there obstacles to lettericity that have bounded γ and unbounded σ?…”