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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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