CHAPTER - 7
GRAPH EQUATIONS FOR LINE GRAPHS,
JUMP GRAPHS, MIDDLE GRAPHS, QLICK
GRAPHS AND PLICK GRAPHS*
*
Part of this chapter has been published in Proc. Nat. Conf. on Graphs, Comb., Algor., and
Appl. (eds S. Arumugam, B. D. Acharya and S. B. Rao) Narosa Ps., New Delhi, India
(2005),
37-40.
ABSTRACT
In this chapter, we solve graph equations
M(G) = Q(H),
L(G) = P(H),
M(G) = P(H),
L(G) = Q(H),
J(G) = Q(H),
M(G) = Q(H), J(G) = P(H) and M(G) = P(H). The equality symbol
‘ = ’ stands for an isomorphism between two graphs.
139
7.1. INTRODUCTION
Cvetkovic and Simic [7] solved graph equations :
L(G) = T(H)
and
L(G) = T(H).
Akiyama, Hamada and Yoshimura [2] solved graph equations :
L(G) = M(H),
M(G) = T(H)
M(G) = T(H),
L(G) = M(H).
Also, they got a solution for the graph equation L(G) = L(H) in
[3]. Further Akiyama, Koneko and Simic [4] obtained solutions for graph
equations L(G) = G\ L(G) = Gn (or L(G) = G7) and L(G) = (g)".
Kulli
and
Patil
[13]
solved
graph
equations
L(G) = TB(H),
L(G) = TB(H), L(G) = Tb(H) and L(G) = T„(H).
In this chapter, we solve completely the following graph equations:
(1)
L(G) = Q(H),
M(G) = Q(H)
(2)
(7)
J(G) = P(H)
(8)
M(G) = P(H).
X
M(G) = Q(H)
ii
(6)
(4)
oT
o
&
J(G) = Q(H),
ii
3
(5)
of
(3)
That is, we obtain all pairs (G, H) of graphs satisfying the
equations (1) to (8).
140
The complement G of a graph G is the graph having the same set
of vertices as G and in which two vertices are adjacent if and only if they
are not adjacent in G.
The jump graph J(G) of a graph G is the graph whose vertex set
coincides with the edge set of G and in which two vertices are adjacent if
and only if they are not adjacent in G. Equivalently, the jump graph J(G)
of G is the complement of line graph L(G) of G [6].
A graph G+ is the endedge graph of a graph G if G+ is obtained
from G by adjoining an endedge u,u' at each vertex u; of G.
For a graph G, let E(G) and b(G) denote its edge set and block set
respectively. If two distinct blocks
Bj
and
Bj
of a graph G are incident
with a common cut-vertex then they are called adjacent blocks of G. The
plick graph P(G) of G is defined as the graph having vertex set
E(G)ub(G), with two vertices adjacent if they correspond to adjacent
edges of G or one corresponds to an edge e{ of G and the other
corresponds to a block Bj of G and
lies in Bj [11]. The qlick graph
Q(G) of G is defined as the graph having vertex set E(G) u b(G), with
two vertices adjacent if they correspond to two adjacent edges of G or to
141
two adjacent blocks of G or one corresponds to an edge e; of G and the
other corresponds to a block Bj of G and ej lies in Bj [12].
We need the following results.
THEOREM 7.A [5]. A graph G is a line graph if and only if none of the
graphs F1? i = 1,2,. . ., 9 of Figure 7.1 is an induced subgraph of G.
THEOREM 7.B [8]. A graph G is the complement of a line graph if and
only if none of the nine graphs Fi, i = 1, 2, . . . , 9 of Figure 7.2 is an
induced subgraph of G.
THEOREM 7.C [9]. Let G be any graph. Then L(G+) = M(G).
THEOREM 7.D [12]. Let G be a connected graph. Then P(G) = Q(G) if
and only if G is a block.
Before investigating solutions of the above graph equations, we
construct a class of graphs which is helpful in our later discussion.
Let G be a graph having cut-vertices such that each block of G is
either C4 or an endbridge. We obtain the new graph by adding a new
edge Xjto each block Bi in such a way that if Bj is C4 then x{ joins
non-cut-vertices of C4 and if Bj is an endbridge then Xj is adjoined at
the pendant vertex of the endbridge. The resulting graph is denoted by Gx.
142
Figure 7.1
143
Figure 7.2
144
7.2. THE SOLUTION OF L(G) = Q(H)
Any graph H which is a solution of the above equation, satisfies the
following properties:
i.
H does not contain the path P4 as an induced subgraph, since
otherwise, F, is an induced subgraph of Q(H).
ii.
H does not contain Cn, n > 5, since otherwise, for n = 5, F9 is
an induced subgraph of Q(H); and for n > 5, F, is an induced
subgraph of Q(H).
iii.
H is not a complete graph Kn, n > 4, since otherwise, F3 is an
induced subgraph of Q(H).
iv.
H is not the complete bipartite graph Kmn, n>3 or m>3,
since otherwise, F, is an induced subgraph of Q(H).
v.
FI does not contain two or more cut-vertices, since otherwise, F,
is an induced subgraph of Q(H).
vi.
If H contains a cut-vertex, then H = K, n, since otherwise, F2 is
an induced subgraph of Q(H).
It follows from the above observations that, H has atmost one
cut-vertex and if H has a cut-vertex, then H = K, n. We consider the
following cases.
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Case 1. Suppose H has exactly one cut-vertex. Then H is K, n, n > 2
and the corresponding graph G is K2 n, n > 2.
Case 2. Suppose H has no cut-vertices. Then H itself is a block. We
consider the following subcases.
Subcase 2.1. H = Kn, n>2. In this case it immediately follows from
observation (iii) that (K, 2, K2) and (K, 4, K3) are the solutions.
Subcase 2.2. H is a complete bipartite graph. It follows from observation
(iv), H is K2 n, n > 2. Since H is a block, G = H + x, where x is the edge
joining the two vertices of degree n in H.
Subcase 2.3. H is neither a complete graph nor a complete bipartite
graph. It follows from observation (ii) that, H is C3, C4 or K4 - x, where
x is any edge of K4. For H = C3or C4, corresponding G is K, 4 or
K4- x respectively. If H = K4- x, then there is no solution because
Q(H) has F3 as an induced subgraph.
Thus we have the following.
THEOREM 7.1. Let G and H be connected graphs. The following pairs
(G, H) are all pairs of graphs satisfying the graph equation L(G) = Q(H):
(Kj, 2, K2); (K2i n, Ki, n) for n > 2; (KM, C3); (K4- x, C4), where x is any
146
edge of K4; and (K2, n + x, K2> n) for n > 2, where x is the edge joining
the two vertices of degree n in K2; n •
7.3 THE SOLUTION OF M(G) = Q(H)
In view of Theorem 7.C, we obtain all pairs of graphs satisfying
the graph equation L(G + ) = Q(H). The complete solutions (G, H) of the
graph equation L(G) = Q(H) are obtained in Theorem 7.1. None of these
solutions is of the form (G+,H). Hence, there is no solution of the
equation M(G) = Q(H).
Hence, we have the next theorem.
THEOREM 7.2. Let G and H be connected graphs. Then there is no
solution of the graph equation M(G) = Q(H).
7.4. THE SOLUTION OF L(G) = P(H)
Let H be any solution of the above equation. It follows from
Theorem 7.D that if H is a block, then Q(H) = P(H) and hence H is a
solution of the graph equation L(G) = P(H).
Suppose H has cut-vertices. Then any block of H is either C4or an
endbridge since otherwise, either Fi or F5 is an induced subgraph of P(H).
Hence H is either K, „, for n > 2 or H is a graph such that any block of H
147
is either C4 or an endbridge. In either ease G = Hx is the corresponding
graph.
Hence the graph equation L(G) = P(H) is solved and we obtain the
following :
THEOREM 7.3. Let G and H be connected graphs. The following pairs
(G, H) are all pairs of graphs satisfying the graph equation L(G) = P(H):
(K12,K2); (K14,C3) ; (K4-x ,C4), where x is any edge of K4;
and (K2n + x ,K2n) for n>2, where x is the edge joining the two
vertices of degree n in K2 n; and (Hx, H) where H is a graph with atleast
one cut-vertex and whose blocks are either C4 or endbridge.
7.5. THE SOLUTION OF M(G) =P(H)
Theorem 7.3 gives complete solutions (G, H) of the graph equation
L(G) = P(H). Among these, none is of the form (G+,H). Hence, from
Theorem 7.C, there is no solution of the equation M(G) = P(H). Thus we
have the following theorem.
THEOREM 7.4. Let G and H be connected graphs. Then there is no
solution of the graph equation M(G) = P(H).
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7.6. THE SOLUTION OF J(G)=Q(H)
Any graph H which is a solution of the above equation satisfies the
following properties:
i.
If H is disconnected, then it has atmost three components, each of
which is K2, since otherwise, F3 is an induced subgraph of Q(H).
ii.
H is not a tree with vertices p> 5, since otherwise, F2 is an
induced subgraph of Q(H).
iii.
H does not contain Cn, n > 7, since otherwise, F2 is an induced
subgraph of Q(H).
iv.
H is not the complete graph Kn, n > 5, since otherwise, Fi is
an induced subgraph of Q(H).
v.
H is not the complete bipartite graph K m n, for m > 4 or n > 4,
since otherwise Fi is an induced subgraph of Q(H).
vi.
H does not contain more than two cut-vertices, since otherwise,
F2 is an induced subgraph of Q(H).
vii.
If H contains a cut-vertex, then every block of H is K2, since
otherwise, Fi is an induced subgraph of Q(H).
Thus H has atmost two cut-vertices and if H has a cut-vertex, then
all its blocks are K 2. We consider the following cases.
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If H has exactly one cut-vertex, then H is K12 or K13.
Corresponding G is 2K12 or C6 respectively.
If H has exactly two cut-vertices. Then H is the path P4.
Corresponding G is the graph shown in Figure 7.3.
If H has no cut-vertices and H is disconnected, then from
observation (i), H is nK2, n < 3 .
For n = l, H = K2 and G = 2K2
For n = 2,H = 2K2 andG=C4
For n = 3, H = 3K2 and G = K4
Now, suppose H has no cut-vertices and H is connected. We
consider the following cases.
Case 1. H = Kn. In this case, it follows from observation (iv) that
(2K2,K2), (4K2,K3) and (3P3 uK2,K4) are the solutions.
Case 2. H = Kmn . Then from observation (v), (C6uK2,K23) and
(K3,
3 uK2 , K33) are the solutions.
Case 3. H is neither a complete graph nor a complete bipartite graph.
From observation (iii), H is Cn, 3 < n < 6 or K4- x, where x is any edge
150
of K4. In this case the solutions are (4K2,C3), (2P3uK2,C4),
(C5uK2,C5), (K2 3 uK2,C6) and (2K, 2 u2K2,K4- x).
In this way, the graph equation is solved and we obtain the
following.
THEOREM 7.5. The following pairs (G, H) are all pairs of graphs
satisfying the graph equation J(G) = Q(H):
(2K, 2,K12); (C6,K, 3); (2K2,K2); (C4,2K2); (K4,3K2); (4K2,K3);
(3P3uK2,K4); (C6uK2,K23); (K3>3uK2,K3t3); (2P3uK2,C4);
(C5
uK2,C5);
(K23uK2,C6); (2K, 2 u2K2,K4- x); and (G,P4) where
G is the graph shown in Figure 7.3.
7.7. THE SOLUTION OF M(G) = Q(H)
It is known that L(G+) = M(G). In Theorem 7.5, we have
investigated the solutions (G, H) of the graph equation J(G) = Q(H).
Among these solutions (2K2,K2) and (4K2,K3) are of the form
(G + ,H). Therefore, the solutions of the graph equation M(G) = Q(H)
are (2K, ,K2) and (4K, ,K3).
151
Figure 7.3
152
From the above discussion, we have the following.
THEOREM 7.6. The following pairs (G, H) are all pairs of graphs
satisfying the graph equation M(G) = Q(H) :
(2K, ,K2) and (4K, ,K3).
7.8. THE SOLUTION OF J(G) =P(H)
In this case, any solution H satisfies the same properties (i) to (vii)
of Section 7.6.
If H is disconnected, then (C4,2K2) and (K4,3K2) are the
solutions.
If H is connected and is a block, then it follows from Case 3 of
Section 7.6 that the solutions are (2K2,K2); (4K2,K3); (3P3 uK2,K4);
(C6 uK2,K2 3); (K33uK2,K33); (2P3uK2,C4); (C5uK2,C5);
(K2 3uK2,C6);and (2K, 2u2K2, K4-x).
If H is connected and has cut-vertices, then it follows from
observations (ii), (vi) and (vii) of Section 7.6, that H is K, 2 or K, 3 or
P4. Then the corresponding graph G is P5 or K3 or a graph shown in
Figure 7.4 respectively.
153
Figure 7.4
154
Thus the graph equation is solved and we have the following.
THEOREM 7.7. The following pairs (G, H) are all pairs of graphs
satisfying the graph equation J(G) = P(H):
<P5,K1>2); (K3+,K, 3); (2K2,K2); (C4,2K2); (K4,3K2); (4K2,K3);
(3P3uK2,K4); (C6uK2,K2>3); (K33 uK2,K3>3); (2P3uK2,C4);
(C5uK2,C5);
(K2> 3 uK2,C6);
(2K, 2 u2K2,K4-x); and (G',P4)
where G' is a graph as shown in Figure 7.4.
7.9. THE SOLUTION OF M(G) = P(H)
The solutions (G, H) of the graph equation J(G) = P(H) are given in
Theorem 7.7. Among these pairs, (K3+,K, 3), (2K2,K2) and (4K2,K3)
are of the form (G+,H). Hence, the solutions of the graph equation
M(G) = P(H) are as follows : (K3,K, 3), (2K,,K2) and (4K,,K3).
Thus we have the next theorem.
THEOREM 7.8. The following pairs (G, H) are all pairs of graphs
satisfying the graph equation M(G) = P(H) :
(K3,Kj 3); (2K,,K2); and(4Kj,K3).
155
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