A geometric graph \G is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call \G a geometric realization of the underlying abstract graph G. A geometric homomorphism from \G to \H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). Geometric homomorphisms can be used to define a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. In this paper, the homomorphism poset of K_{3,3} is determined.
Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.
Let ${\cal H}(P,C_3)$ be the set of order homomorphisms from a poset $P$ to the chain $C_3 = 1 < 2 < 3$. We develop a recursive approach for the calculation of the cardinality of ${\cal H}(P,C_3)$, and we apply it on several types of posets, including $P = C_3 \times C_3 \times C_k$ and $P = {\cal H}(C_k, C_3)$; for the latter poset $P$, we derive a direct formula for $\# {\cal H} ( P, C_3 )$.
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 $P \in \mathfrak{P}_r$, a one-to-one mapping $ρ_P : {\cal H}(P,R) \rightarrow {\cal H}(P,S)$ exists which fulfills a certain regularity condition. It is shown that $R \sqsubseteq_G S$ is equivalent to $\# {\cal S}(P,R) \leq \# {\cal S}(P,S)$ for every finite posets $P$, where ${\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. In consequence…
We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on finite products of chains. For the special case of a Boolean algebra, we observe that the corresponding homomorphism complex is isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this work can be interpreted as a generalization of the study of these complexes. We provide a detailed investigation when our poset is a product of…
Let $\mathrm{XY_{L,T}}$ denote the class of countably infinite $L$-structures that satisfy the axioms $T$ and in which all homomorphisms of type X (these could be homomorphisms, monomorphisms, or isomorphisms) between finite substructures of $M$ are restrictions of an endomorphism of $M$ of type Y (for example, an automorphism or a surjective endomorphism). Lockett and Truss introduced 18 such morphism-extension classes for relational structures. For a given pair $L,T$, however, two or more morphism-extension properties may define the same class of structures. In this paper, we establish all e…
Given finite posets $P$ and $Q$, we consider a specific ideal $L(P,Q)$, whose minimal monomial generators correspond to order-preserving maps $ϕ:P\rightarrow Q$. We study algebraic invariants of those ideals. In particular, sharp lower and upper bounds for the Castelnuovo-Mumford regularity and the projective dimension are provided. Precise formulas are obtained for a large subclass of these ideals. Moreover, we provide complete characterizations for several algebraic properties of $L(P,Q)$, including being Buchsbaum, Cohen-Macaulay, Gorenstein and having a linear resolution. We also give a pa…
A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $ϕ$ of $\cal S$ such that $ϕ\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.
A signed graph $(G, σ)$ is a graph $G$ along with a function $σ: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph $(G, σ)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, π)$ to which $(G, σ)$ admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention …
We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct unlabelled edge-subgraphs. We study these posets given up to isomorphism, and their relation to the reconstruction conjectures. We show that when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be constructed from each other. The result implies that trees are reconstructible from their abstract bond lattice. We present many results on the reconst…
A homomorphism of a signed graph $(G, σ)$ to $(H, π)$ is a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ such that adjacencies, incidences and the product of signs of closed walks are preserved. Motivated by reformulations of the $k$-coloring problem in this language, and specially in connection with results on $3$-coloring of planar graphs, such as Grötzsch's theorem, in this work we consider bounds on maximum average degree which are sufficient for mapping to the signed graph $(K_{2k}, σ_m)$ ($k\geq 3$) where $σ_m$ assigns to edges of a perfect matching the…
We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an $R$-labeling, imply the existence of the (non-homogeneous) ${\bf cd}$-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existe…
We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source…
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $\Hom$ complexes in terms of spaces of equivariant poset maps and $Γ$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $…
In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a different…
We consider a database composed of a set of conceptual graphs. Using conceptual graphs and graph homomorphism it is possible to build a basic query-answering mechanism based on semantic search. Graph homomorphism defines a partial order over conceptual graphs. Since graph homomorphism checking is an NP-Complete problem, the main requirement for database organizing and managing algorithms is to reduce the number of homomorphism checks. Searching is a basic operation for database manipulating problems. We consider the problem of searching for an element in a partially ordered set. The goal is to…
