List homomorphisms to separable signed graphs

Minimum cost and list homomorphisms to semicomplete digraphs

It has been attracting much attention to make use of list homomorphisms in parallel programming because they ideally suit the divide-and-conquer parallel paradigm. However, they have been usually treated rather informally and ad hoc in the development of efficient parallel programs. What is worse is that some interesting functions, e.g., the maximum segment sum problem, are basically not list homomorphisms. In this article, we propose a systematic and formal way for the construction of a list homomorphism for a given problem so that an efficient parallel program is derived. We show, with several well-known but nontrivial problems, how a straightforward, and “obviously” correct, but quite inefficient solution to the problem can be successfully turned into a semantically equivalent “almost list homomorphism.” The derivation is based on two transformations, namely tupling and fusion, which are defined according to the specific recursive structures of list homomorphisms.

The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees [1] and reflexive signed graphs [18]. Irreflexive signed graphs are the heart of the problem, and Kim and Siggers have formulated a conjectured classification for these signed graphs. We focus on a special case of irreflexive signed graphs, namely those in which the unicoloured edges form a spanning path or cycle, and classify the complexity of list homomorphisms to these signed graphs. In particular, our results confirm the conjecture of Kim and Siggers for this class of signed graphs.

In the deletion version of the list homomorphism problem, we are given graphs G and H, a list $$L(v)\subseteq V(H)$$ for each vertex $$v\in V(G)$$ , and an integer k. The task is to decide whether there exists a set $$W \subseteq V(G)$$ of size at most k such that there is a homomorphism from $$G {\setminus } W$$ to H respecting the lists. We show that DL-Hom( $${H}$$ ), parameterized by k and |H|, is fixed-parameter tractable for any $$(P_6,C_6)$$ -free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom( $${H}$$ ) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. (Combinatorica 19(4):487–505, 1999), a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

Extension problems with degree bounds

The structural information contentIg(X) of a graphX was treated in detail in three previous papers (Mowshowitz 1968a, 1968b, 1968c). Those investigations ofIg point up the desirability of defining and examining other entropy-like measures on graphs. To this end the chromatic information contentIc(X)of a graphX is defined as the minimum entropy over all finite probability schemes constructed from chromatic decompositions having rank equal to the chromatic number ofX. Graph-theoretic results concerning chromatic number are used to establish basic properties ofIc on arbitrary graphs. Moreover, the behavior ofIc on certain special classes of graphs is examined. The peculiar structural characteristics of a graph on which the respective behaviors of the entropy-like measuresIc andIg depend are also discussed.

This chapter provides an overview of the diachronic development of the left periphery in German. It introduces the OV/V2 asymmetry as a basic property of continental West Germanic syntax, as well as the components of the verb-second rule. On this basis, it surveys the rise of verb-second, elaborating on state-of-the-art in the beginning of the attestation, on the relation between V2 and the emergence of complementizers in Germanic, as well as on the role of Germanic sentence particles in the left periphery of the clause. In addition, orders challenging the validity V2 in German—such as verb-first, verb-third, and verb-final orders—are discussed. The chapter also discusses the role of information structure in movement to the left periphery, as well as the emergence of a special class of adverbial connectives, which develop from low adverbs and acquire a special status with respect to the left periphery.

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.