For graphs G and H , a homomorphism from G to H is a function φ : V (G) → V (H), which maps vertices adjacent in G to adjacent vertices of H . A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in H . Many cases of graph homomorphism and locally injective graph homomorphism are NPcomplete, so there is little hope to design polynomial-time...

We define the Homomorphism Extension (HomExt) problem: given a group G, a subgroup M ≤ G and a homomorphism φ : M → H , decide whether or not there exists a homomorphism φ̃ : G → H extending φ, i.e., φ̃|M = φ. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem i...

We study the complexity of structurally restricted homomorphism and constraint satisfaction problems. For every class of relational structures C, let LHOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B, when each element in A is only allowed a certain subset of elements of B as its image. We prove, under a certain complexity-theore...

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...

Bounds on the range of random graph homomorphism into Z, and the maximal height difference of the Gaussian random field, are presented.

Eilenberg’s variety theorem gives a bijective correspondence between varieties of languages and varieties of finite semigroups. The second author gave a similar relation between conjunctive varieties of languages and varieties of semiring homomorphisms. In this paper, we add a third component to this result by considering varieties of meet automata. We consider three significant classes of lang...

0.1. Endomorphisms of elliptic curves. Recall that a homomorphism of complex elliptic curves is just a holomorphic map E1 → E2 which preserves the origin. (It turns out that this condition is enough to force it to be a homomorphism of groups in the usual sense; why?) An isogeny of elliptic curves is a homomorphism whose kernel is a finite subgroup of E1. In fact the kernel of a homomorphism of ...

We show that for every binary matroid $N$ there is a graph $H_*$ such the graphic $M_G$ of $G$, matroid-homomorphism from to if and only graph-homomorphism $G$ $H_*$. With this we prove complexity dichotomy problem $\rm{Hom}_\mathbb{M}(N)$ deciding $M$ admits homomorphism $N$. The polynomial time solvable has loop or no circuits odd length, otherwise $\rm{NP}$-complete. also get dichotomies lis...