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graph homomorphism (Definition)

We shall define what a graph homomorphism is between two pseudographs, so that a graph homomorphism between two multigraphs, or two graphs are special cases.

Recall that a pseudograph $ G$ is an ordered triple $ (V,E,i)$, where $ V$ is a set called the vertex set of $ G$, $ E$ is a set called the edge set of $ G$, and $ i:E\to 2^V$ is a map (incidence map), such that for every $ e\in E$, $ 1\le \vert i(e)\vert\le 2$.

Elements of $ V$ are called vertices, elements of $ E$ edges. The vertex/vertices associated with any edge $ e$ via the map $ i$ is/are called the endpoint(s) of $ e$. If an edge $ e$ has only one endpoint, it is called a loop. $ e_1$ and $ e_2$ are said to be parallel if $ i(e_1)=i(e_2)$.

As two examples, a multigraph is a pseudograph such that $ \vert i(e)\vert=2$ for each edge $ e\in E$ (no loops allowed). A graph is a multigraph such that $ i$ is one-to-one (no parallel edges allowed).

Definition. Given two pseudographs $ G_1=(V_1,E_1,i_1)$ and $ G_2=(V_2,E_2,i_2)$, a graph homomorphism $ h$ from $ G_1$ to $ G_2$ consists of two functions $ f:V_1\to V_2$ and $ g:E_1\to E_2$, such that

$\displaystyle i_2\circ g=f^*\circ i_1.$     (1)

The function $ f^*:2^{V_1} \to 2^{V_2}$ is defined as $ f^*(S)=\lbrace f(s)\mid s\in S\rbrace$.

A graph isomorphism $ h=(f,g)$ is a graph homomorphism such that both $ f$ and $ g$ are bijections. It is a graph automorphism if $ G_1=G_2$.

Remarks.

  • In case when $ G_1$ and $ G_2$ are graphs, a graph homomorphism may be defined in terms of a single function $ f:V_1\to V_2$ satisfying the condition (*)
    $ \lbrace v_1,v_2\rbrace$ is an edge of $ G_1\Longrightarrow \lbrace f(v_1),f(v_2)\rbrace$ is an edge of $ G_2$.
    To see this, suppose $ h=(f,g)$ is a graph homomorphism from $ G_1$ to $ G_2$. Then $ f^*i_1(e)=f^*(\lbrace v_1,v_2\rbrace)=\lbrace f(v_1),f(v_2)\rbrace=i_2(g(e))$. The last equation says that $ f(v_1)$ and $ f(v_2)$ are endpoints of $ g(e)$. Conversely, suppose $ f:V_1\to V_2$ is a function sastisfying condition (*). For each $ e\in E_1$ with end points $ \lbrace v_1,v_2\rbrace$, let $ e^{\prime}\in E_2$ be the edge whose endpoints are $ \lbrace f(v_1),f(v_2)\rbrace$. There is only one such edge because $ G_2$ is a graph, having no parallel edges. Define $ g:E_1\to E_2$ by $ g(e)=e^{\prime}$. Then $ g$ is a well-defined function which also satisfies Equation (1). So $ h:=(f,g)$ is a graph homomorphism $ G_1\to G_2$.
  • The definition of a graph homomorphism between pseudographs can be analogously applied to one between directed pseudographs. Since the incidence map $ i$ now maps each edge to an ordered pair of vertices, a graph homomorphism thus defined will respect the “direction” of each edge. For example,
    $\displaystyle \begin{xy} *!C\xybox{ \xymatrix{a \ar[r] & b\ar[d] \ c \ar[u] &... ...uad\qquad\qquad \xymatrix{r \ar[r] & s \ t \ar[u]\ar[r] & u\ar[u]} } \end{xy}$
    are two non-isomorphic digraphs whose underlying graphs are isomorphic. Note that one digraph is strongly connected, while the other is only weakly connected.



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See Also: graph isomorphism, pseudograph, multigraph, graph

Other names:  graph automorphism
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Cross-references: strongly connected, digraphs, ordered pair, well-defined, end points, equation, terms, bijections, graph isomorphism, functions, one-to-one, parallel, loop, endpoint, vertices, map, edge, vertex, graphs, multigraphs, pseudographs
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This is version 8 of graph homomorphism, born on 2006-07-13, modified 2007-05-24.
Object id is 8140, canonical name is GraphHomomorphism.
Accessed 2127 times total.

Classification:
AMS MSC: 05C60 (Combinatorics :: Graph theory :: Isomorphism problems )
 05C75 (Combinatorics :: Graph theory :: Structural characterization of types of graphs)
Pending Errata and Addenda
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