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Factoring Cartesian-product graphs. (English) Zbl 0811.05054
The authors consider the structure of the equivalence relation $$\sigma$$ on the edge set of a graph $$G$$ belonging to the decomposition of $$G$$ into a Cartesian product of prime graphs; this prime factorization theory was founded by G. Sabidussi [Math. Z. 72, 446-457 (1960; Zbl 0093.376)]. For a family of graphs $$(G_ \iota)_{\iota \in I}$$, the Cartesian product $$\Gamma = \square_{\iota \in I} G_ \iota$$ is the graph $$\Gamma = (V(\Gamma), E(\Gamma))$$ with $$V(\Gamma) := \{(x_ \iota)_{\iota \in I} : x_ \iota \in V(G_ \iota)$$, $$\iota \in I\}$$ and $$E(\Gamma) := \bigcup_{\kappa \in I} E_ \kappa$$, where $$E_ \kappa := \{\{(x_ \iota), (y_ \iota)\} : (x_ \iota), (y_ \iota) \in V(\Gamma) \wedge \{x_ \kappa, y_ \kappa\} \in E(G_ \kappa) \wedge \forall \iota (\iota \in I - \{\kappa\} \to x_ \iota = y_ \iota)\}$$, $$\kappa \in I$$; for any $$a = (a_ \iota) \in V(\Gamma)$$, the connected component $$G$$ of $$\Gamma$$ containing $$a$$ is called the weak Cartesian product $$G = \square^ a_{\iota \in I} G_ \iota$$. The decomposition $$E(G) = \bigcup_{\kappa \in I}(E_ \kappa \cap E(G))$$ determines an equivalence relation $$\sigma(\square^ a G_ \iota)$$ on $$E(G)$$ belonging to this weak Cartesian product. For any connected graph $$G$$, it is defined (a) $$e, f \in E(G)$$ are in the relation $$\delta$$ iff either $$e$$ and $$f$$ are adjacent and there is no chordless square containing $$e$$ and $$f$$ or $$e = f$$ or $$e$$ and $$f$$ are opposite edges of a chordless square; (b) $$e = \{x, y\} \in E(G)$$ and $$f = \{x', y'\} \in E(G)$$ are in (Djoković’s) relation $$\Theta$$ iff the distances satisfy $$d(x,x') + d(y,y') \neq d(x,y') + d(x', y)$$; (c) a subgraph $$H$$ of $$G$$ is convex (in $$G$$) iff every shortest $$G$$-path between any two vertices of $$H$$ is in $$H$$; (d) an equivalence relation $$\gamma$$ on $$E(G)$$ with the equivalence classes $$E_ \iota$$, $$\iota \in I$$, is convex iff for any $$K \subseteq I$$ every connected component of the subgraph induced on $$\bigcup_{\iota \in K} E_ \iota$$ is convex; (e) $$e = \{x, z\} \in E(G)$$ and $$f = \{z, y\} \in E(G)$$ are in the relation $$\tau$$ iff $$z$$ is the unique common neighbour of $$x$$ and $$y$$. Obviously, for any weak Cartesian product representation $$\square^ a _{\iota \in I} G_ \iota$$ of $$G$$ the relation $$\sigma(\square^ a G_ \iota)$$ is convex and contains the transitive closure $$\delta^*$$ of $$\delta$$.
Now, the following main results are proved for any (finite or infinite) connected graph $$G$$: (1) Every convex equivalence relation $$\gamma \supseteq \delta$$ on $$E(G)$$ induces a representation of $$G$$ as a weak Cartesian product. (2) The intersection of an arbitrary set of convex equivalence relations on $$E(G)$$ containing $$\delta$$ is convex. Thus there is a finest convex equivalence relation on $$E(G)$$ containing $$\delta$$, the convex hull $$C(\delta)$$ of $$\delta$$. (3) From (1) and (2) it follows a new proof of the known statement that $$G$$ has a unique representation as a weak Cartesian product of prime graphs (prime factorization). (4) The equivalence relation $$\sigma$$ on $$E(G)$$ belonging to this prime factorization is $$\sigma = C(\delta)$$. (5) $$\sigma = (\delta \cup \Theta)^*$$; moreover, every convex equivalence relation $$\gamma \supseteq \tau$$ on $$E(G)$$ contains $$\delta$$, and $$\sigma = (\tau \cup \Theta)^* = C(\tau)$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C99 Graph theory 05C12 Distance in graphs
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