On subgraphs of Cartesian product graphs and S-primeness

On subgraphs of Cartesian product graphs and S-primeness

On subgraphs of Cartesian product graphs and S-primeness Bostjan Bresar, University of Maribor, FK, Vrbanska 30, 2000 Maribor, e-mail: bostjan.bres...

On subgraphs of Cartesian product graphs and S-primeness

Bostjan Bresar, University of Maribor, FK, Vrbanska 30, 2000 Maribor, e-mail: [email protected]

Abstract We consider S-prime graphs, i.e. graphs which cannot be represented as a nontrivial subgraphs of a Cartesian product of two graphs. A characterization of S-prime graphs with connectivity 2 is given, providing a substantialy larger class of examples of such graphs as they were previously known. We then introduce a class of socalled simple S-prime graphs and a concept of extension of graphs, and prove a characterization of S-prime graphs via these concepts.

Key words:

Cartesian product, prime, subgraph.

We consider undirected graphs without loops or multiple edges. The Cartesian product G1

2

has as a vertex set whenever

u

=

x

of graphs G1 = (V (G1 ); E (G1 ) and G2 = (V (G2 ); E (G2 ))

G2 V

and

(G 1 )

vy

2



V

(G2 ), and vertices (u; v ); (x; y ) are adjacent

E (G2 );

or

ux

2

called S-composite if there exist graphs of

G1

2

G2 ,

S-prime

but

G

E (G 1 )

G1 ; G2

is not a subgraph of any of

and

=

y.

A graph

such that

G

is a subgraph

G1 ; G 2 .

(with respect to the Cartesian product) if

G

v

A graph

G

G

is

is called

is not S-composite. The

problem of describing the structure of S-prime graphs can be considered as one of the basic problems of graph products, and is close to the problem of characterizing prime graphs (in fact any S-prime graph is also prime, while, for example,

P3

is prime and not S-prime). We mention that the equivalent

problem (of S-prime graphs) for the other three standard graph products has relatively straightforward solutions. The subgraphs of Cartesian products and S-prime graphs were investigated by Lamprey and Barnes [4,5]. They provided a characterization of S-prime graphs via so-called basic S-prime graphs, which turned out to be precisely S-prime graphs (on at least three vertices) that include no proper S-prime graphs (on at least three vertices). They proved that any S-prime graph can

Preprint submitted to Elsevier Preprint

29 May 2000

be obtained from basic S-prime graphs by two operations. Recently, Klavzar et al.  proved a characterization of S-prime graphs which we shall also use. A surjective mapping c : V (G) ! f1; 2; : : : ; kg is called a k-coloring of G (not to be confused with the usual coloring of graphs). Let c be a k-coloring of G and let P be a path of G on at least three vertices. Then we say that P is well-colored if for any two consecutive vertices u and v of P we have c(u) 6= c(v ). We call a k-coloring c of G a path k -coloring if for any wellcolored path of G we have c(u) 6= c(v ) for all pairs of vertices u; v in the path.

Theorem 1  Let G be a connected graph on at least three vertices. Then G is S-composite if and only if there exists a path k-coloring of G with 2 k  jV (G)j 1:



Although the above result does not reveal the structure of S-prime graphs, it gives a new insight in these graphs. We will use it in the following de nition of graphs, which are S-composite, but are, in a way, close to be S-prime.

De nition 2 A graph G is called (S-composite) u; v -critical if G is S-

composite and G has two special vertices u; v such that for any path k-coloring c of G we have (A) if c(u) = c(v ) then c(x) = c(u) for all x 2 V (G); (B ) if c(u) 6= c(v ) then there exists (B:1) a well-colored path from u to a vertex of color c(v )

and (B:2) a well-colored path from v to a vertex of color c(u).

G is called (S-composite) weakly u; v -critical if we change the above de nition in (B), so that (B.1) or (B.2) are true, and G is not u; v -critical. A u; v -critical graph is called basic u; v -critical if it does not contain any u; v critical subgraphs. If a weakly u; v -critical graph, in addition, does not contain any weakly u; v -critical subgraphs, then it is called basic weakly u; v -critical graph. Using these de nitions we are able to prove a characterization of basic S-prime graphs with connectivity 2. A cut set C of G is a set of vertices in G such that G C is a disconnected graph. We denote by X1 ; X2 ; : : : ; Xn the connected components of G C , and we call G1 ; G2 : : : ; Gn ; induced by vertex subsets V (Xi ) [ C; parts of G C . A connectivity (G) of a graph G is the minimum cardinality of any cutset of G. 2

Theorem 3

u

Let G be a graph with connectivity 2, having a cut set of vertices and v . Then G is a basic S-prime graph if and only if

(i) G{fu; v g has two parts, one a basic basic weakly u; v -critical graph, or

u; v -critical

graph, and the other a

(ii) two basic u; v -critical parts which have no weakly or (iii) it has three basic weakly u; v -critical parts.

u; v -critical

subgraphs,

The problem of searching basic S-prime graphs is thus transformed to two similar problems, namely searching basic u; v -critical and basic weakly u; v critical graphs. However, these two classes are in close connection, which can be deduced from the above theorem. Namely, by gluing two weakly u; v -critical graphs in vertices u and v , we obtain a u; v -critical graph. Another possible constructions is shown in the next theorem.

Theorem 4

A graph

(basic) weakly critical.

G

with a vertex

u; w-critical

w

if and only if

of degree

G{fwg

1,

having a neighbor

is (respectively basic)

v , is u; v -

Combining Theorem 3 with Theorem 4 one can construct several new in nite classes of basic S-prime graphs. In particular, one can nd basic S-prime graphs with diameter as large as one wants, which has not yet been known. Our main result uses the concepts of simple S-prime graphs and extensions. Similar concepts are often used in graph theory, especially when graph products are involved (e.g., median and quasi-median graphs [1,6,7] as well as isometric subgraphs of Hamming graphs and isometric subgraphs of hypercubes , which use so-called expansion and/or amalgamation procedures). In the process of extension we replace (locally) a part of a graph with another part, so that certain properties of graphs are preserved.

De nition 5 if one of the

3

(i) A basic S-prime graph cases occur:

-

G

is

-

G

is isomorphic to

G is called a simple S-prime graph

3-connected

K2 3 ;

(G) = 2 and for any cut set of G with vertices a; b, G{fa; bg has two parts, one being isomorphic to P3 :

-

(ii) A basic called a if

u1 ; u2 -critical

(respectively basic weakly

a; b-critical)

graph

G

is

simple u; v -critical graph (respectively simple weakly u; v -critical)

(P) for any cut set of vertices

a; b;

and any component

3

X

of

G{fa; bg

which

has none of the vertices ui

(i = 1; 2)

f

g

we have: if H is a part of G{ a; b

corrresponding to X; and is weakly a; b-critical, then H is isomorphic to P3 .

De nition 6

0

A graph G is obtained from G by an

extension if G has a cut

set of vertices u; v such that one of its parts is P3 ; and G

0

is obtained from

G by replacing this part with any simple weakly u; v -critical graph or a simple u; v -critical graph without weakly u; v critical subgraphs.

Theorem 7

A graph G is a basic S-prime graph if and only if it can be

obtained from a simple S-prime graph by a sequence of extensions.

The theorem above extends the characterization of Lamprey and Barnes, which bases on the de niton of basic S-prime graphs . Simple S-prime graphs form a building blocks, together with simple weakly u; v -critical graphs (and possibly simple u; v -critical graphs which have no weakly u; v -critical subgraphs), from which one can obtain any basic S-prime graph, hence any Sprime graph. At present we know of 5 simple S-prime graphs, and 21 simple weakly u; v -critical graphs. Since the problem of a (nice) structural characterization of these two classes of graphs seems to be quite diÆcult, some new examples of these graphs would already be helpful.

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