On subgraphs of Cartesian product graphs and Sprimeness
Bostjan Bresar, University of Maribor, FK, Vrbanska 30, 2000 Maribor, email:
[email protected]
Abstract We consider Sprime graphs, i.e. graphs which cannot be represented as a nontrivial subgraphs of a Cartesian product of two graphs. A characterization of Sprime 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 Sprime graphs and a concept of extension of graphs, and prove a characterization of Sprime 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 Scomposite if there exist graphs of
G1
2
G2 ,
Sprime
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 Scomposite. The
problem of describing the structure of Sprime 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 Sprime graph is also prime, while, for example,
P3
is prime and not Sprime). We mention that the equivalent
problem (of Sprime graphs) for the other three standard graph products has relatively straightforward solutions. The subgraphs of Cartesian products and Sprime graphs were investigated by Lamprey and Barnes [4,5]. They provided a characterization of Sprime graphs via socalled basic Sprime graphs, which turned out to be precisely Sprime graphs (on at least three vertices) that include no proper Sprime graphs (on at least three vertices). They proved that any Sprime graph can
Preprint submitted to Elsevier Preprint
29 May 2000
be obtained from basic Sprime graphs by two operations. Recently, Klavzar et al. [3] proved a characterization of Sprime graphs which we shall also use. A surjective mapping c : V (G) ! f1; 2; : : : ; kg is called a kcoloring of G (not to be confused with the usual coloring of graphs). Let c be a kcoloring of G and let P be a path of G on at least three vertices. Then we say that P is wellcolored if for any two consecutive vertices u and v of P we have c(u) 6= c(v ). We call a kcoloring 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 [3] Let G be a connected graph on at least three vertices. Then G is Scomposite if and only if there exists a path kcoloring of G with 2 k jV (G)j 1:
Although the above result does not reveal the structure of Sprime graphs, it gives a new insight in these graphs. We will use it in the following de nition of graphs, which are Scomposite, but are, in a way, close to be Sprime.
De nition 2 A graph G is called (Scomposite) u; v critical if G is S
composite and G has two special vertices u; v such that for any path kcoloring 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 wellcolored path from u to a vertex of color c(v )
and (B:2) a wellcolored path from v to a vertex of color c(u).
G is called (Scomposite) 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 Sprime 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 Sprime 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 Sprime 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; wcritical
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 Sprime graphs. In particular, one can nd basic Sprime graphs with diameter as large as one wants, which has not yet been known. Our main result uses the concepts of simple Sprime graphs and extensions. Similar concepts are often used in graph theory, especially when graph products are involved (e.g., median and quasimedian graphs [1,6,7] as well as isometric subgraphs of Hamming graphs and isometric subgraphs of hypercubes [2], which use socalled 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 Sprime graph cases occur:

G
is

G
is isomorphic to
G is called a simple Sprime graph
3connected
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; bcritical)
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; bcritical, 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 Sprime graph if and only if it can be
obtained from a simple Sprime graph by a sequence of extensions.
The theorem above extends the characterization of Lamprey and Barnes, which bases on the de niton of basic Sprime graphs [5]. Simple Sprime 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 Sprime graph, hence any Sprime graph. At present we know of 5 simple Sprime 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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