I Introduction
Graphical passwords (GPWs) are familiar with people in nowadays, such as 1dimension code, 2dimension code, face authentication, fingerprint authentication, speaking authentication, and so on, in which 2dimension code is widely used in everywhere of the world. A 2dimension code can be considered as a GPW, since it is a picture. Researchers have worked on GPWs for a long time ([5, 6, 7]). Wang et al. propose another type of graphic passwords (Topsnutgpws) in [20] and [21], which differ from the existing GPWs.
As an example, we have two Topsnutgpws shown in Fig.1(a) and (b), where is as a public key, is as a private key. The authentication in network communication is given in Fig.1(c). By observing Fig.1 carefully, we can see that the labels of nodes (also, vertices) and edges of two Topsnutgpws and form a complementary relationship, and the labels of each edge and its two nodes in and satisfies some certain mathematical restraints. Another important character of Topsnutgpws is the configuration, also, the topological structure (called graph hereafter). Thereby, we say that Topsnutgpws are naturalinspired from mathematics of view. In general, Topsnutgpws are easy saved in computer by algebraic matrices, and Topsnutgpws occupy small space rather than that of the existing GPWs such that Topsnutgpws can be implemented quickly.
Topsnutgpw can be as a platform for password, cipher code and encryption of information security. As Topsnutgpws were made by “topological configurations plus number theory”, we will apply a particular class of matrices to describe Topsnutgpws for the purpose of writing easily in computer and running quickly by computer. These matrices are called Topsnutmatrices, and can yield randomly textbased passwords (TBpaws for short) for authentication and encryption in communication. For the theoretical base, we will introduce some operations on Topsnutmatrices in order to implement them for building up TBpaws flexibly.
As known, Topsnutgpws are related with many mathematical conjectures or NPproblems, so Topsnutgpws are computationally unbreakable or provable security. A Topsnutgpw has an advantage, that is, it can generate textbased passwords with longer byte such that it is impossible to rebuild the original Topsnutgpw from the derivative textbased passwords made by . This derives us to explore the area of generating textbased passwords from Topsnutgpws in this article. We believe this transformation from Topsnutgpws to textbased passwords is very important for the real application of Topsnutgpws.
Ia Examples and problems
We write “textbased passwords” by TBpaws, and “topological graphic passwords” as Topsnutgpws hereafter, for the purpose of quick statement. We will make some TBpaws from a Topsnutgpw depicted in Fig.2. Along a path shown in Fig.2, we have a TBpaw
obtained from the labels of vertices and edges on the path .
The Topsnutgpw depicted in Fig.2
admits an oddelegant labelling
such that each edge holds to be an odd number, and for any pair of vertices , as well as for any two edges and of . By cryptography of view, the Topsnutgpw has twelve subTopsnutgpws and with to form a larger authentication, where with are public keys, and with are private keys. Moreover, a subTopsnutgpw pictured in Fig.3 distributes us a TBpawObviously, to reconstruct the subTopsnutgpw from the TBpaw is difficult, and the TBpaw does not rebuild the original Topsnutgpw at all. It means that the procedure of generating TBpaws from Topsnutgpws is irreversible. On the other hands, this Topsnutgpw can distribute us TBpaws shown in the formula (22), such that each TBpaw has at least 380 bytes or more.
For the encryption of data and dynamic networks, we propose the following problems:

How to generate TBpaws from a given Topsnutgpw?

How many TBpaws with the desired byte are there in a given Topsnutgpw?

How to encrypt a dynamic network by Topsnutgpws or TBpaws?
We will try to find some ways for answering partly the above problems in the later sections. In graph theory, Topsnutgpws are called labelled graphs, so both concepts of Topsnutgpws and labelled graphs will be used indiscriminately in this article.
IB Preliminary
The following terminology, notation, labellings, particular graphs and definitions will be used in the later discussions.

The notation indicates a consecutive set with integers holding , denotes an oddset with odd integers with respect to , and is an evenset with even integers .

The number of elements of a set is written as .

is the set of vertices adjacent with a vertex , is called the degree of the vertex . If we call a leaf.

A lobster is a tree such that the deletion of leaves of the tree results in a caterpillar, where the deletion of leaves of a caterpillar produces just a path.

A graph having vertices and edges is called a graph.

A spider is a tree having paths with and , its own vertex set , such that its own edge set , and . Clearly, , and for any vertex . We call as the body, and each path is a leg of length of .

A ringlike network has a unique cycle , and each vertex of is coincident with some vertex of a tree with .

The set of all subsets of a set is denoted as , but the empty set is not allowed in . For example, for a set , then contains: , , , , , , , , , , , , , , .
Definition 1.
[33] A labelling of a graph is a mapping such that for any pair of elements of , and write the label set . A dual labelling of a labelling is defined as: for . Moreover, is called the vertex label set if , the edge label set if , and a universal label set if .
A combinatoric definition of setlabellings is as follows.
Definition 2.
[33] Let be a graph.
(i) A set mapping is called a total setlabelling of if for distinct elements .
(ii) A vertex set mapping is called a vertex setlabelling of if for distinct vertices .
(iii) An edge set mapping is called an edge setlabelling of if for distinct edges .
(iv) A vertex set mapping and a proper edge mapping are called a vset eproper labelling of if for distinct vertices and two edge labels for distinct edges .
(v) An edge set mapping and a proper vertex mapping are called an eset vproper labelling of if for distinct edges and two vertex labels for distinct vertices .
Definition 3.
([4, 34, 41]) Suppose that a connected graph with admits a mapping . For edges the induced edge labels are defined as . Write , . There are the following restrictions:

.

.

, .

, .

.

.

is a bipartite graph with the bipartition such that ( for short).

is a tree containing a perfect matching such that for each edge .

is a tree having a perfect matching such that for each edge .
A graceful labelling holds (a), (c) and (e) true; a setordered graceful labelling satisfies (a), (c), (e) and (g), simultaneously; a strongly graceful labelling holds (a), (c), (e) and (h) true; a strongly setordered graceful labelling complies with (a), (c), (e), (g) and (h) meanwhile. An oddgraceful labelling holds (a), (d) and (f) true; a setordered oddgraceful labelling obeys (a), (d), (f) and (g), simultaneously; a strongly oddgraceful labelling holds (a), (d), (f) and (i) true at the same time; a strongly setordered oddgraceful labelling fulfils (a), (d), (f), (g) and (i), simultaneously.
Another group of definitions is about the sum of end labels of edges, we present it as follows:
Definition 4.
([4, 42]) A graph with admits a labelling , where is an integer set. For edges the induced edge labels are defined as or for every edge . And is the vertex label set, and is the edge label set. There are the following constraints:

.

.

.

.

.

.

when is even, and when is odd.

.

.

.

.

.

.

.

.

There exists an integer so that .

is bipartite with its bipartition so that .
We call to be: (1) a felicitous labelling if c3, c8 and c10 hold true; (2) an oddelegant labelling if c4, c9 and c13 hold true; (3) a harmonious labelling if c2, c8 and c10 hold true, when is a tree, exactly one edge label may be used on two vertices; (4) a properly even harmonious labeling if c5, c9 and c11 hold true; (5) a harmonious labeling if c2, c6 and c15 hold true; (6) an even sequential harmonious labeling if c5, c7 and c12 hold true; (7) a harmonious harmonious labeling if c1, c6 and c14 hold true; (8) a strongly harmonious labeling if c3, c8, 16 and c10 hold true; (9) a setordered harmonious labeling if c3, c8, c17 and c10 hold true; (10) an setordered oddelegant labelling if c4, c9, c17 and c13 hold true;
Ii Techniques for generating TBpaws from Topsnutgpws
Our methods for generating TBpaws from Topsnutgpws are mainly based on the following disciplines: Topsnutconfigurations, graphlabellings, Topsnutmatrices, Topsnutmatchings and graphic groups, these are two invariable quantities of Topsnutgpws.
Iia Topsnutconfigurations
By simple and clear statements, we utilize the oddgraceful/oddgraceful labellings and Topsnutconfiguration to show several methods for creating TBpaws.
IiA1 Pathneighbormethod
is a caterpillar of a graph admitting a setordered oddgraceful labelling . So, the deletion of leaves of is just a path in the caterpillar , such that each has its own leaf set with and , and the vertex set is
See a caterpillar depicted in Fig.4. Thereby, we can get a vvtype TBpaw and a vevtype TBpaw
by the pathmethod for deriving two types of TBpaws from Topsnutgpws.
From a path revealed in Fig.5, we can get a vvtype TBpaw and a vevtype TBpaw by the pathmethod.
Next, we introduce the pathneighbormethod.
Let a vertex have its neighbor set with , we have a vvtype TBpaw
and
by the miniprinciple, and moreover we get a vvtype TBpaw
and another vevtype TBpaw
by the maxiprinciple. Let with , where . By the miniprinciple, for the edge , we write a vvtype TBpaw
(1) 
by the miniprinciple, denoted as
(2) 
and moreover we can write a vevtype TBpaw
(3) 
by the miniprinciple, denoted as
(4) 
Similarly with (2) and (4), we can write and by the maxiprinciple.
For example, by means of a caterpillar exhibited in Fig.5 and two formulae (1) and (3), we have two vvtype TBpaws
according to the miniprinciple and the maxiprinciple. Similarly,
is a vevtype TBpaw by the miniprinciple, and moreover,
is obtained by the maxiprinciple.
It is easy to see that there are many ways to generate vvtype/vevtype TBpaws from a Topsnutgpw made by a labelled caterpillar, except the miniprinciple and the maxiprinciple. In a vvtype/vevtype TBpaw , we say , , and in the vvtype/vevtype TBpaw . So, we have permutations for writing , and a caterpillar with the path distributes us vvtype/vevtype TBpaws at least.
IiA2 Cycleneighbormethod
By a caterpillar depicted in Fig.4, we add an edge to for joining the vertex with , the resulting graph is denoted as , in which there is a cycle . So, we have a vvtype TBpaw
(5) 
along the cycle , and a vevtype TBpaw
(6) 
Since we have initial vertices of the cycle , so the number of vvtype/vevtype TBpaws distributed from is equal to
(7) 
IiA3 Lobsterneighbormethod
In [41] and [42], the authors have proven: Each lobster admits one of oddgraceful labelling and oddelegant labelling. Thereby, we can apply lobsters to make Topsnutgpws, or we select subTopsnutgpws being lobsters of Topsnutgpws to derive vvtype/vevtype TBpaws. Another advantage about lobsters is helpful for us to produce random Topsnutgpws that generate random vvtype/vevtype TBpaws.
Recall, a lobster is defined as a tree such that the deletion of leaves of results in a caterpillar, that is, the remainder is just a caterpillar, where is the set of all leaves of . In other words, each lobster can be constructed by adding leaves to some caterpillar. The results in [41] and [42] enable us to build up lobsters admitting oddgraceful/oddelegant labellings by caterpillars admitting setordered oddgraceful/oddelegant labellings through adding leaves.
We show an example for illustrating “adding leaves to a caterpillar admitting a setordered oddgraceful labelling produces a lobster admitting an oddgraceful labelling”. Based on a caterpillar , as revealed in Fig.5, we can see that Fig.6 gives the procedure of “adding randomly leaves to ”, and the labelling new edges is shown in Fig.7, and moreover the procedure of “labelling new vertices and relabelling old vertices” presents the desired oddgraceful lobster (see Fig.8).
We, now, come to introduce the lobsterneighbormethod for getting vvtype/vevtype TBpaws from a Topsnutgpw made by an oddgraceful lobster in the following algorithm.
Theorem 1.
There exists an efficient and polynomial algorithm (LOBSTERalgorithm) for generating vvtype/vevtype TBpaws from Topsnutgpws made by oddgraceful lobsters.
Proof.
We, directly, use an algorithmic proof here for generating vvtype/vevtype TBpaws from Topsnutgpws.
Step 1. Suppose that a lobster corresponds to a caterpillar obtained by deleting some leaves from . Write as the set of deleted leaves, so . Conversely, is obtained by adding the leaves of to . Let be the path as the remainder after the deletion of leaves of the caterpillar , and let be a setordered oddgraceful labelling of . Thereby, we have
with and
with . Thereby, we have a vvtype TBpaw
(8) 
and a vevtype TBpaw
(9) 
Step 2. Adding randomly leaves to for forming a lobster . Since is a setordered oddgraceful labelling of the caterpillar , so with , and any edge of holds and such that . By the hypothesis above, we can write with for , and with for . Suppose that each vertex is added leaves from the set with , and each vertex is added leaves from the set with . Here, it is allowed some or . The resulting tree is just . Therefore,
and write .
We define a labelling for in the following steps.
Substep 2.1. We label the edges of in the increasing order: for , for , and
with . Thus, .
Substep 2.2. For the edges , we set in the decreasing order: with , with , and
with and .
Substep 2.3. We come to label the vertices of in the following way: for ; for with ; for ; and for with
Step 3. Producing a vvtype TBpaw and a vevtype TBpaw from the lobster . We use the notation to denote the set of new leaves added to hereafter. We set for with , and get a vvtype subTBpaw
with , where is the set of new leaves added to and for . Hence, we get the desired vvtype TBpaw
(10) 
Next, for getting a vevtype TBpaw from the lobster , we take
for , and moreover
for . So,
with . Thereby, the lobster distributes a vevtype TBpaw as follows
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