Lexicographic product of graphs

In graph theory, the lexicographic product or graph composition $G ∙ H$ of graphs $G$ and $H$ is a graph such that

the vertex set of $G ∙ H$ is the cartesian product $V(G) \times V(H)$ ; and
any two vertices $(u, v)$ and $(x, y)$ are adjacent in $G ∙ H$ if and only if either $u$ is adjacent with $x$ in $G$ or $u = x$ and $v$ is adjacent with $y$ in $H$ .

If the edge relations of the two graphs are order relations, then the edge relation of their lexicographic product is the corresponding lexicographic order.

The lexicographic product was first studied by Felix Hausdorff (1914). As Feigenbaum & Schäffer (1986) showed, the problem of recognizing whether a graph is a lexicographic product is equivalent in complexity to the graph isomorphism problem.

Properties

The lexicographic product is in general noncommutative: $G ∙ H \neq H ∙ G$ . However it satisfies a distributive law with respect to disjoint union: $(A + B) ∙ C = A ∙ C + B ∙ C$ . In addition it satisfies an identity with respect to complementation: $C(G ∙ H) = C(G) ∙ C(H)$ .

The independence number of a lexicographic product may be easily calculated from that of its factors (Geller & Stahl 1975):

α(G ∙ H) = α(G)α(H)

The clique number of a lexicographic product is as well multiplicative:

ω(G ∙ H) = ω(G)ω(H)

References

Feigenbaum, J.; Schäffer, A. A. (1986), "Recognizing composite graphs is equivalent to testing graph isomorphism", SIAM Journal on Computing 15 (2): 619–627, doi:10.1137/0215045, MR 0837609 .
Geller, D.; Stahl, S. (1975), "The chromatic number and other functions of the lexicographic product", Journal of Combinatorial Theory, Series B 19: 87–95, doi:10.1016/0095-8956(75)90076-3, MR 0392645 .
Hausdorff, F. (1914), Grundzüge der Mengenlehre, Leipzig
Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN 0-471-37039-8

External links

Weisstein, Eric W., "Graph Lexicographic Product" from MathWorld.