IJMS, Vol. 11, No. 3-4, (July-December 2012), pp. 309-318
© Serials Publications
ISSN: 0972-754X
SELF-ASSEMBLY OF GRAPHS ON EDGES
R. Rama, Suresh Badarla, Kalpana Mahalingam &
Suresh Kumar Yadav
Abstract: Self-assembly is a process in which simple objects autonomously combine
themselves into larger objects. Two simple connected graphs can be self assembled on
edges. The decomposition of possibly self-assembled graphs given a set of generators of
the given graph. A decomposition algorithm is given for this purpose which leads to a
method of finding HPP in a given graph. If the given graph (possibly self-assembled) has
HPP, then the base has HPP. Converse need not be true.
Keywords: Edge-self-assembly, Base, k-trees, Proper decomposition.
1. INTRODUCTION
The process in which relatively simple components brought into contact with each other
experience local interactions guided by basic rules, and combine to form increasingly
complex structures is called self-assembly. There is no externally guiding the force or
direction, just the summation of the undirected local interactions. The progress in science
has experienced many of the naturally occurring self-assembling systems. Also progress in
technology has made attempts to mimic nature and create artificial self-assembling systems.
There is substantial development in creating vastly powerful computing systems,
nano-biomedical devices etc. Scientists have been pursuing various approaches to studying
self-assembly [4, 7, 9, 10]. Some of the topics of such research are Self-assembly in the tile
assembly model, Programming algorithmic self-assembly, DNA nano technology and
Self-assembly data structures like strings. Formal languages, central to understanding
computation of the strings, provide a natural formalism for DNA based computing. Several
other frameworks inspired by (linear) self-assembly exists see [1, 2, 4, 5]. The model
proposed in [9] concentrates on two-dimensional self-assembly of DNA complexes; linear
self-assembly comes as a particular case of this model. The model proposed in [5, 6, 8]
uses graphs (two-dimensional) self-assembly.
This paper deals with self-assembly on edges of graphs. The iterated self-assembly
produces a family of graphs. We also give a method of finding a generator set of graphs for
two given (possibly) self-assembled graphs. This approach throws interest on exploring
the presence HPP in a given graph.
1st International Conference on Mathematics and Mathematical Sciences (ICMMS), 7 July 2012.
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R. Rama, Suresh Badarla, Kalpana Mahalingam & Suresh Kumar Yadav
2. SELF-ASSEMBLY OF GRAPHS ON EDGES
We follow the terminologies and the basic notations of graph theory as in [3]. A simple
graph G over V is an ordered pair G = (V, E) where V is the finite set of vertices (or nodes),
E is finite set of edges of the form (u, v), u, v � V, u � v, where each edge is an unordered
pair of vertices. An edge (u, v) means that one end-point of the edge is the vertex u and the
other end-point is the vertex v. Edge set of G is written as E (G) and the vertex set of G as
V (G). The number of vertices of graph G is called the order of the graph (written as O (G))
and the number of edges of the graph is called the size of the graph. We consider only
simple graphs where repeated edges (multiple edges) with same end-points and edges with
end-points same (loops) are not allowed. The neighbourhood of a vertex v in G is the set of
vertices adjacent to v (written as NG (v)). The set of all simple graphs of all possible orders
and sizes is denoted by . By contained in , we mean the set of graphs contained in .
For � , | | denotes the number of graphs in . A k-tree is either a complete graph on
(k + 1) vertices or given a k-tree G� with n vertices, a k-tree G with (n + 1) vertices can be
constructed by introducing a new vertex v and picking a k-clique Q in G� and then joining
each vertex u in Q to the new vertex v.
Definition 2.1.1: Let G1 and G2 be any two simple connected graphs in . Then
edge-self-assembly of G1 and G2 is denoted as EDSe (G1, G2) and defied as follows:
EDSe (G1, G2) = {G : G = G1 > e < G2,
where G1(e) = G2(e) = e}.
Let G1, G2 be two simple connected graphs then by self-assembling G1, G2 on edge we
mean, we super impose the two graphs on the edge retaining the remaining portion of
G1, G2 intact.
For example if the graph G1 :=
and G2 :=
Hence self-assembly of G1 and G2 will be
G :=
Here G is one of the possible outputs of self-assembly. There are 42 possible outputs of
self-assembling G1 and G2 on edges and some of them may be isomorphic.
Remark: The set of all possible graphs G got by edge-self-assembly of two graphs
G1 and G2 is denoted by EDSe (G1, G2). That is G is got by super imposition of an edge
Self-Assembly of Graphs on Edges
311
of G1 with an edge of G2. If G � EDSe (G1, G2), then | V (G) | = | V (G1) | + | V (G2) | – 2
and | E (G) | = | E (G1) | + | E (G2) | – 1.
1,
Definition 2.1.2: For any two set of graphs
EDSe ( 1,
2
)�
�
G1 �
G2 �
2
� ,
ESDSe (G1 , G2 )
1
2
We write EDSe ( ) instead of EDSe ( , ). We also define EDSen ( ), an iterated version
of EDSe, for the set of graphs as EDSen ( ) = EDSe ( , EDSen – 1 ( )), n � 1. Where EDSe0
( ) = . Accordingly, the set of graphs got by the edge-self-assembly of the graphs in is
EDSe+ ( ) = n � 1 EDSen ( ) .
Examples: 1. If
= {K2}, then EDSe+ (L) = {K2}.
2. If = {K3}, then EDSe ( ) = EDSe (K3, K3) is a 2-tree on 4 vertices and EDSen ( )
will be a 2-tree on (n + 3) vertices, n � 1.
3. Let G1 = P3 and G2 = P4 then EDSe (G1, G2) = EDSe (P3, P4)
= {P5 and
4. If
}
= {Pl, l � 3}, then EDSe+ (Pl, Pm) will be a tree of (l + m) – 2 vertices.
Definition 2.1.3: We say that a set of graphs is closed with respect to iterated edge
self assembly operation EDSe+, if EDSe+ ( ) � . The following closure properties are
observed with respect to EDSe+.
1. The set of cycle graphs is not closed
2. The set of complete graphs is not closed
3. The set of 2-trees is closed
4. The set of Chordal graph is closed
Theorem 2.1.1: If has graphs of order greater than or equal to two and all graphs in
are of same order, then is not closed with respect to iterated edge-self-assembly.
Theorem 2.1.2: For any simple graphs G and H, we have EDSe (G, H) = EDSe (H, G)
and EDSe (G, H) � Ø, for any non trivial graphs G and H.
Theorem 2.1.3: All the graphs in EDSe (G, G) are not isomorphic.
Definition 2.1.4: Consider the following sequence of graphs EDSASEQ (G) =
{G, EDSe (G), EDSe2 (G), ...}. Let us call this sequence as edge-self-assembly sequence
of graph G. Every element in this sequence is a set of graphs.
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R. Rama, Suresh Badarla, Kalpana Mahalingam & Suresh Kumar Yadav
A graph H is said to be reachable from G if H � EDSASEQ (G). EDSASEQ (G) is said to
be saturated if EDSASEQ (G) contains only finite number of elements. That is edge-self-assembly
may not produce any more new member.
Examples: 1. Any 2-tree is reachable from G = K3.
2. Let G = K2, then EDSASEQ (G) = {K2} is saturated.
Theorem 2.1.4: EDSASEQ (G) will have a minimum of one element and no element in
the sequence is empty.
Proof: The first element of EDSASEQ (G) will be G itself and second element is
EDSe (G, G) � Ø. In the (n + 1)th element of EDSen (G) every graph is got by edge-selfassembling the graphs from EDSen – 1 (G) with G, by induction, we can note that no element
in the sequence is empty.
Theorem 2.1.5: For any non-trivial graph G (� K2), EDSASEQ (G) is never saturated.
Theorem 2.1.6: If H is reachable from a non-trivial graph G (� K2), then H cannot be
isomorphic to G.
2.2 Decomposition Algorithm
In this section, we study the existence of a minimal base for a given set of graphs, and if
exists how to construct the same. We use the term ‘minimal’ in the sense, without all the
elements of the base. One cannot construct the desired set of graphs by the edge-selfassembly process. For this, we define the concept of decomposition of graphs into smaller
graphs. Decomposition of a graph into smaller graphs is what a set of graphs is to base.
Lemma 2.2.1: For any set of graphs any graphs L � Y, any graph G � EDSe+( ), there
exist graphs {G1, G2, ..., Gn} � and edges {e1, e2, ..., en – 1} such that G = G1 > e1 < G2 >
e2 < G3 ... Gn – 1 > en – 1 < Gn.
Proof: G � EDSe+ ( ) means that G is in EDSe ( ) or EDSe2 ( ) or EDSe3 ( ) etc. Suppose
G � EDSen – 1 ( ). i.e., G � EDSe ( , EDSen – 2 ( )). There exists G1 � and G�2 � EDSen – 2 ( )
such that G = G1 > e1 < G�2, where e1 is the self-assembly edge. Since G�2 � EDSen – 2 ( ),
there exists G2 � and G�3 � EDSen – 3 ( ) such that G�2 = G2 > e2 < G�3. continuing this, we
have G�n – 1 � EDSe ( ), there exists Gn – 1, and Gn � such that G�n – 1 = Gn – 1 > en – 1 < Gn.
Combining all these, we have G = G1 > e1 < G2 > e2 < G3 > e3 < ... Gn – 1 > en – 1 < Gn.
By the above Lemma 2.2.1, we call {G1, G2, ..., Gn} as an -decomposition of G
and write that G = EDSe (G1, G2, ..., Gn), which means that G has -decomoposition. In an
-decompostion of G, all the elements of graphs G need not be unique.
Lemma 2.2.2: For any
-decomposition.
, EDS e+ ( ) is the set of all graphs with non trivial
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Lemma 2.2.3: An -decomposition of a graph is order independent, in the sense that
EDSe (G1, G2, ..., Gn) Gi’s, i = 1, ..., n can be placed in any order.
Definition 2.2.1: Let � be any set of graphs. A graph G is said to be (locally)
irreducible over
with respect to the edge-self-assembly operation if G � EDSe+ \ .
Otherwise G is composite.
Definition 2.2.2: (a) Graphs G � which cannot be obtained by edge-self-assembly
of other members of , then we say that G has trivial decomposition.
(b) A graph G is said to be globally irreducible (GI) if G � EDSe+ ( )\ for any
or G � .
�
Lemma 2.2.4: Let be any set of graphs. A graph G has no -decomposition iff G is
locally irreducible over .
Proof: Given that G has no -decomposition. This means that, there exists no graph
Gi � , i = 1, 2, ..., n such that G = EDS (G1, G2, ..., Gn). This implies that G cannot be
obtained by edge-self-assembly of some graphs of . That means, G � EDSe+ ( )\ . Hence,
G is locally irreducible over . The other part of the proof is got by retracing the above
steps.
Lemma 2.2.5: Let be any set of graphs. If G has only one -decomposition such that
G = EDS (G1, G2, ..., Gn), then Gi , i = 1, 2, ..., n are locally irreducible over .
Proof: Let G = EDS (G1, G2, ..., Gn) is the only one -decomposition. This implies that
no graph Gi , i = 1, 2, ..., n can be -decomposed further. That means Gi s are in and Gi s
have trivial decomposition. This implies that Gi s all are irreducible over .
Definition 2.2.3: Let , B be any two set of graphs. B is said to be a base of if
� EDSe+ (B). B is said to be minimal base for , if for any other base B� of , B� � B
implies that B� = B.
Now, we state a sequence of Lemmas to answer the questions like, whether a base
exists for any set of graphs and is so, how to construct them.
Lemma 2.2.6: Let
irreducible over .
Lemma 2.2.7: If
be any set of graphs. All graphs in a minimal base of
has a base, then it has a unique minimal base.
Note: Unique minimal base of a set of graphs
Lemma 2.2.8: For any set of graphs
over .
Theorem 2.2.1:
irreducible graphs.
must be
,
EDSe+
will have all the irreducible graphs.
( ) is the set of all composite graphs
will have a minimal base if and only if
does not have any globally
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R. Rama, Suresh Badarla, Kalpana Mahalingam & Suresh Kumar Yadav
Proof: It is clear that if have a minimal base, then does not have any globally
irreducible graphs. Suppose does not have any global irreducible graphs in it, then there
exists a composite graph G which is locally irreducible over . Hence \{G} is a base.
Continuing like this for any composite graph, we can obtain minimal base for .
The above results are indicating that graphs can be decomposed into say {G1, G2, ..., Gn}
such that each Gi is globally irreducible graph. Since globally irreducible graphs cannot be
reduced further, we say that such a decomposition of G will be unique for G. We call such
decomposition {G1, G2, ..., Gn} as proper decomposition of G. Thus, we have the following
Lemma.
Lemma 2.2.9: Proper decomposition of a graph is unique.
Remark: Cliques are globally irreducible.
Theorem 2.2.2: Let G be any graph, suppose G contains a clique of maximum size k
and also contain a vertex of degree greater than or equal to k, then G is composite otherwise
G is globally irreducible.
Theorem 2.2.3: Let be set of graphs with no globally irreducible graphs. Then the
base of , B = G � {Gi : Gi � Proper decomposition}.
Decomposition Algorithm:
• Stage 1: Given a composite graph G, if G is a complete graph, then stop.
• Stage 2: Let [v1, v2, ..., vn] be the vertices of the graph G.
° Step 1: for i = 1 to n
find out NG (vi)
for i = 1 to n – 1
for j = i + 1 to n
find NG (vi)
NG (vj)
for i = 1 to n – 1
for j = i + 1 to n
if ((| NG (vi) NG (vj) | = maximum or zero) && (vi, vj) � E (G)) then choose the
edge (vi, vj) is a self-assembled edge.
° Step 2: Let e = (vi, vj) and V = V1 V2 such that V1 V2 = {vi, vj} and G1 is the
induced sub graph of G with vertex set V1 and G2 is the induced sub graph of G
with vertex set V2.
Self-Assembly of Graphs on Edges
315
° Step 3: if either G1 = G or G2 = G, then choose a different edge.
• Stage 3: The decomposed graphs G1 and G2 should be such that G = G1 > e < G2.
• Stage 4: If the vertex set V (G) cannot be written as the union of V1 and V2 as above,
then the graph G is globally irreducible. Then, we conclude G in the set of proper
decomposition of G.
• We repeat the stage 2, 3, and 4 for G1 and G2.
• The algorithm terminates when there is no graph to be decomposed to further. When
the algorithm terminates, proper decomposition of G is output.
Example: Consider the graph G and the vertex set of G be V = {v1, v2, v3, v4, v5, v6, v7, v8, v9}
Now we will find out the neighbourhood of every vertex in the graph. NG (v1) = {v4, v5,
v6, v7}, NG (v2) = {v3, v6, v8, v9}, NG (v3) = {v2, v8, v9}, NG (v4) = {v1, v5, v7}, NG (v5) = {v1, v4, v6},
NG (v6) = {v1, v2, v5}, NG (v7) = {v1, v4}, NG (v8) = {v2, v3}, NG (v9) = {v2, v3}.
Next we will find the intersection of neighbourhoods of each pair of adjacent vertices.
NG (v1) NG (v4) = {v5, v7}, NG (v1) NG (v5) = {v4, v6}, NG (v1) NG (v6) = {v5}, NG (v1)
NG (v7) = {v4}, NG (v2) NG (v3) = {v8, v9}, NG (v2) NG (v6) = Ø.
NG (v2)
NG (v8) = {v3}, NG (v2)
NG (v9) = {v3}, NG (v3)
NG (v8) = {v2}
NG (v3)
NG (v9) = {v2}, NG (v4)
NG (v5) = {v1}, NG (v4)
NG (v7) = {v1}
NG (v5)
NG (v6) = {v1}.
We will find out the maximum or zero cardinality of the intersections of neighbourhoods
of adjacent vertices. | NG (v1) NG (v4) | = 2, | NG (v1) NG (v5) | = 2, | NG (v2) NG (v3) | = 2,
| NG (v2) NG (v6) | = 0.
Here we can choose any one among the edges {(v1, v4), (v1, v5), (v2, v3), (v2, v6)} as a
self-assembled edge.
Step 2: Suppose the edge e = (v1, v4) as a self-assembled edge of G and let V = V1 V2
where V1 = {v1, v4, v7} and V2 = {v1, v2, v3, v4, v5, v6, v7, v8, v9} such that V1 V2 = {v1, v4}.
Let G1 be the induced sub graph G of with vertex set V1 and G2 be the induced sub
graph of G with vertex set V2 then
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R. Rama, Suresh Badarla, Kalpana Mahalingam & Suresh Kumar Yadav
v9
and
Stage 3: The decomposed graphs are G1 and G2 should be such that G = G1 > e < G2
on the edge (v1, v4).
Further decomposition of G1 is not possible. The decomposition of G2 as follows:
Step 2: Let the edge e = (v1, v5) as a self-assembled edge of G2 and let V = V1 V2
where V1 = {v1, v4, v5} and V2 = {v1, v2, v3, v4, v5, v6, v7, v8, v9} such that V1 V2 = {v1, v5}.
Let G3 be the induced sub graph of G2 with vertex set V1 and G4 be the induced sub
graph of G2 with vertex set V2 then
Stage 3: The decomposed graphs are G3 and G4 should be such that G2 = G3 > e < G4
on the edge (v1, v5).
Further decomposition of G3 is not possible. The decomposition of G4 as follows:
Step 2: Let the edge e = (v2, v3) as a self-assembled edge of G4 and let V = V1
where V1 = {v1, v2, v3, v5, v6, v9} and V2 = {v2, v3, v8} such that V1 V2 = {v2, v3}.
V2
Let G5 be the induced sub graph of G4 with vertex set V1 and G6 be the induced sub
graph of G4 with vertex set V2 then
Stage 3: The decomposed graphs are G5 and G6 should be such that G4 = G5 > e < G6
on the edge (v2, v3).
Further decomposition of G6 is not possible. The decomposition of G5 as follows:
Self-Assembly of Graphs on Edges
317
Step 2: Let the edge e = (v2, v6) as a self-assembled edge of G5 and let V = V1
where V1 = {v1, v2, v5, v6} and V2 = {v2, v3, v6, v9} such that V1 V2 = {v2, v6}.
V2
Let G7 be the induced sub graph of G5 with vertex set V1 and G8 be the induced sub
graph of G5 with vertex set V2 then
Stage 3: The decomposed graphs are G7 and G8 should be such that G5 = G7 > e < G8
on the edge (v2, v6).
The decomposition of G7 as follows:
Step 2: Let the edge e = (v1, v6) as a self-assembled edge of G7 and let V = V1
where V1 = {v1, v2, v6} and V2 = {v1, v5, v6} such that V1 V2 = {v1, v6}.
V2
Let G9 be the induced sub graph of G7 with vertex set V1 and G10 be the induced sub
graph of G7 with vertex set V2 then
Stage 3: The decomposed graphs are G9 and G10 should be such that G7 = G9 > e < G10
on the edge (v1, v6).
The decomposition of G8 is similar to the decomposition of G7 and the further
decomposition of G9 and G10 are not possible. Therefore the proper decomposition of G
contains {P3, K3} only.
3. CONCLUDING REMARKS
Self-assembly of graphs on edges is new approach of looking at graphs for its sub graphs.
The induced sub graphs forming the base set are the irreducible sub graphs. Since cliques
are irreducible, the decomposition algorithm identifies all the cliques in the given graph.
We also observe that if the given graph has HPP, the base set of graphs (each one) has HPP,
but the converse need not be true. Looking at the above said points, one can make new
characterisation of graphs in terms of possessing HPP in them.
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R. Rama, Suresh Badarla, Kalpana Mahalingam &
Suresh Kumar Yadav
Department of Mathematics,
Indian Institute of Technology,
Chennai, India.
E-mails: [email protected],
2
[email protected],
3
[email protected], & [email protected]