In this paper, we introduce a path embedding problem inspired by the well-known hydrophobic-polar (HP) model of protein folding. A graph is said bicolored if each vertex is assigned a label in the set {red, blue}. For a given bicolored path P and a given bicolored graph G, our problem asks whether we can embed P into G in such a way as to match the colors of the vertices. In our model, G represents a protein's “blueprint,” and P is an amino acid sequence that has to be folded to form (part of) G. We first show that the bicolored path embedding problem is NP-complete even if G is a rectangular grid (a typical scenario in protein folding models) and P and G have the same number of vertices. By contrast, we prove that the problem becomes tractable if the height of the rectangular grid G is constant, even if the length of P is independent of G. Our proof is constructive: we give a polynomial-time algorithm that computes an embedding (or reports that no embedding exists), which implies that the problem is in XP when parameterized according to the height of G. Additionally, we show that the problem of embedding P into a rectangular grid G in such a way as to maximize the number of red-red contacts is NP-hard. (This problem is directly inspired by the HP model of protein folding; it was previously known to be NP-hard if G is not given, and P can be embedded in any way on a grid.) Finally, we show that, given a bicolored graph G, the problem of constructing a path P that embeds in G maximizing red-red contacts is Poly-APX-hard.

The paper deals with the shortest path-based in-trees on a grid graph. There a root is supposed to move among all vertices. As such a spanning mobility pattern, root trajectories based on a Hamilton path or cycle are discussed. Along such a trajectory, each vertex randomly selects the next hop on the shortest path to the root. Under those assumptions, this paper shows that a root trajectory termed an S-path provides the minimum expected symmetric difference. Numerical experiments show that another trajectory termed a Right-cycle also provides the minimum result.

Vigna and Ghezzi showed that the language of grid graphs could not be constructed by their context-free graph grammars [1]. In this paper, we construct a context-sensitive graph grammar for the two-dimensional grid graphs.