This paper unifies problems and results related to (embedding) universal and homomorphism universal structures. On the one side we give a new combinatorial proof of the existence of universal objects for homomorphism defined classes of structures (thus reproving a result of Cherlin, Shelah and Shi) and on the other side this leads to the new proof of the existence of dual objects (established by Nešetřil and Tardif). Our explicite approach has further applications to special structures such as variants of the rational Urysohn space. We also solve a related extremal problem which shows the optimality (of the used lifted arities) of our construction (and a related problem of A. Atserias).
A partial order is universal if it contains every countable partial order as a suborder. In 2017, Fiala, Hubička, Long and Nešetřil showed that every interval in the homomorphism order of graphs is universal, with the only exception being the trivial gap $[K_1,K_2]$. We consider the homomorphism order restricted to the class of oriented paths and trees. We show that every interval between two oriented paths or oriented trees of height at least 4 is universal. The exceptional intervals coincide for oriented paths and trees and are contained in the class of oriented paths of height at most 3, wh…
An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a morphism $φ:V(G)\rightarrow V(H)$ such that each edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation). An $(m,n)$-colored-mixed graph $T$ is said to be $P_g^{(m, n)}$-universal if every graph in $P_g^{(m, n)}$ (the planar $(m, n)$-colored-mi…
RIKEN Center for Advanced Intelligence Project（日本理化学研究所先进智能项目）；Tokyo Institute of Technology（东京工业大学）
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some family of graphs (e.g. paths or non-isomorphic trees). We show that graph homomorphism numbers provide a natural invariant (isomorphism invariant and $\mathcal{F}$-invariant) embedding maps which can be used for graph classification. Viewing the expressive power of a graph classifier by the $\mathcal{F}$-indistinguishable concept, we prove the universality pro…
