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Finite digraphs $R$ and $S$ are studied with $\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for every finite digraph $G \in \mathfrak{ D }'$, where ${\cal H}(G,H)$ is the set of order homomorphisms from $G$ to $H$ and $\mathfrak{ D }'$ is a class…

Let $\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \sqsubseteq_G S$ iff, for every $P…

For digraphs $G$ and $H$, let ${\cal H}(G,H)$ be the set of all homomorphisms from $G$ to $H$, and let ${\cal S}(G,H)$ be the subset of those homomorphisms mapping all proper arcs in $G$ to proper arcs in $H$. From an earlier investigation…

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives…

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Ne\v{s}et\v{r}il introduced a relaxed version of…

Graph homomorphism has been an important research topic since its introduction [17]. Stated in the language of binary relational structures in that paper [17], Lov\'asz proved a fundamental theorem that, for a graph $H$ given by its $0$-$1$…

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the…

A large driver of the complexity of graph learning is the interplay between structure and features. When analyzing the expressivity of graph neural networks, however, existing approaches ignore features in favor of structure, making it…

A relational structure is homomorphism-homogeneous (HH-homogeneous for short) if every homomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. Similarly, a…

A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where…

We introduce (weak) oddomorphisms of graphs which are homomorphisms with additional constraints based on parity. These maps turn out to have interesting properties (e.g., they preserve planarity), particularly in relation to homomorphism…

Let the join of two graphs be the union of two disjoint graphs connected by $j$ edges in a one-to-one manner. In previous work by Gyurov and Pinzon, which generalized the results of Badura and Rara, the determinant of the adjacency matrix…

We consider a generalization of finding a homomorphism from an input digraph $G$ to a fixed digraph $H$, HOM($H$). In this setting, we are given an input digraph $G$ together with a list function from $G$ to $2^H$. The goal is to find a…

We introduce pathographs as a framework to study graph classes defined by forbidden structures, including forbidding induced subgraphs, minors, etc. Pathographs approximately generalize s-graphs of L\'ev\^eque--Lin--Maffray--Trotignon by…

For a fixed graph H, the H-Recoloring problem asks whether for two given homomorphisms from a graph G to H, we can transform one into the other by changing the image of a single vertex of G in each step and maintaining a homomorphism from G…

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce…

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency…

Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping…

Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$…

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general…