On the Total Vertex Irregular Labeling of Proper Interval Graphs

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers).  For a simple graph G = (V, E) with vertex set V and edge set E, a labeling  Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling  Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is deﬁned to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which  a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.


Introduction
In graph theory, the field of graph labeling is growing fast during the last three decades. Labeled graphs has wide range of applications including coding theory, x-ray, crystallography, radar, astronomy, circuit design, channel assignments of FM radio stations and communication network addressing. A labeling of a graph is a function from the vertex set V or the edge set E or both to the set of natural numbers subject to certain conditions. If the domain is the vertex-set (edge-set) the labeling is called vertex-labeling (edge labeling). For a graph G = (V, E) a labeling : V ∪ E → {1, 2, ..., k} is said to be a vertex irregular total k-labeling of the graph G if for every two different vertices x and y of G, wt(x)≠ wt(y) where the weight of a vertex x in the labeling is where N (x) is the set of neighbors of x. The minimum k for which the graph G has an vertex irregular total k-labeling is called the total vertex irregularity strength of the graph G and is denoted by tvs(G). The irregularity strength s(G) of a graph G can be defined only for graphs containing at most one isolated vertex and no connected component of order 2. On the other hand, the total vertex irregularity strength tvs(G) is defined for every graph G. This paper dedicated to the study of total vertex irregularity strength of a subclass of interval graphs, called proper interval graphs.
An undirected graph G =(V, E) is an interval graph, if the vertex set V can be put into one to one correspondence with a set of intervals I on the real line R such that two vertices are adjacent in G, if and only if their corresponding intervals have non empty intersection. A graph G is a proper interval graph, if there is an interval representation of G in which no interval properly contains another. The intervals and the vertices of an interval graph are very similar things. Interval graphs are discussed extensively in [1]. Here, we assume that the input graph is given by an interval representation I which is the set of n sorted intervals labeled by 1, 2, . . . , n.
Let I={I 1 , I 2 , …, I n } where I j =[a j , b j ], j =1, 2, 3, . . . , n; be the interval representation of the given interval graph G =(V, E), V ={1, 2, . . . , n}, a j and b j are the left and the right end points of the interval I j . Without any loss of generality, we assume that each interval contains both its end points and that no two intervals share a common end point. Also, we assume that the intervals in I are indexed by increasing right end points, that is, b 1 < b 2 < · · · < b n . This indexing is known as IG ordering. Fig. 1 shows an interval graph and its corresponding interval representation. An interval graph can be recognized in O(n + m) time which yields an interval representation in the certain instance. All graphs considered in this paper are connected, undirected and have no self loop or parallel edges.
To date, the total vertex irregularity strengths are known only for some special classes of graphs. Bača et al. determined exact value of tvs for stars, cliques and prisms [5]. Also, they fixed boundary value of tvs for tree and (p, q) graph.
If G is a (p, q) graph with minimum degree δ and maximum degree ∆, then Bača et al. [5] proved that These results were then improved by Przybylo [14] for sparse graphs and for graphs with large minimum degree. Anholcer et al. established a new upper bound of the form [9] ( ) Among the others, Ahmad et al. [6] fixed the lower bound of tvs of any graph. Moreover, Ahmad et al. [7,8] found an exact value of the total vertex irregularity strength for Jahangir graphs, circulant graphs and wheel related graphs. Wijaya and Slamin determined the exact values of the total vertex irregularity strength of wheels, fans, suns and friendship graphs [10]. Furthermore, Wijaya et al. found the total vertex irregularity strength for complete bipartite graphs [11]. Recently, a restricted variation of total irregular labeling is appeared in [15].
In this study we find the TVS of proper interval graphs, a subclass of interval graphs and present an algorithm to determine the TVS of proper interval graphs. Moreover, the boundary values of tvs of proper interval graphs and interval graphs are presented.

Notations and Preliminaries
A clique of a graph is a set of vertices, such that, there exists an edge between every pair of vertices in the set. Lemma 1. (Fulkerson and Gross [16]). A triangulated graph and so an interval graph with n vertices has at most n maximal cliques. The number of maximal cliques is n if and only if the graph has no edges.
A connected interval graph with n vertices has at most n − 1 maximal cliques, when the graph is a path. Gilmore and Hoffman have shown that the maximal cliques of an interval graph G can be linearly ordered such that for every vertex x ∈ V the maximal cliques containing x occur consecutively [17]. Theorem 1. Let G be an undirected graph. The following statements are equivalent [17].
(i) G is an interval graph.
(ii) G does not contain chordless cycle with four or more vertices and its complement ̅ is a comparability graph.
(iii) The maximal cliques of G can be linearly ordered such that for every vertex x of G, the maximal cliques containing x occur consecutively.
Then weight of the vertices v 1 , v 2 , . . . , v n are n + 1, n + 2, . . . , 2n respectively. Therefore ϕ is a vertex irregular total labeling function and C n has vertex irregular total labeling. Clearly, 2 is the minimum integer such that ϕ : V ∪ E → {1, 2} is a vertex irregular total labeling function. Therefore tvs(C n ) = 2.
The degree of all vertices of an interval graph G can be obtained in O(n 2 ) time. Let D r be the set of all vertices of G of degree r. Then the vertices of G can be partitioned into the sets D δ , D δ+1 , . . ., D ∆ , where δ and ∆ are the minimum and maximum degree of G respectively. Now, we prove a property of proper interval graph which plays an important role in our algorithm. Lemma 3. The number of leaf vertex of a connected proper interval graph is at most 2. Proof: Let G be a connected proper interval graph with n vertices 1, 2, . . . , n. Therefore none of the intervals properly contain any other interval. Then the intervals 2, 3, . . . , n − 1 must intersect at least two intervals. This implies that the degree of each of the internal vertices 2, 3, . . . , n − 1 must be greater than 1. Therefore, only the end vertices 1 and n are potential leaf vertex.
From this lemma, it is observed that the cardinality of the set D 1 is 0 or 1 or 2, that is, |D 1 | ≤ 2. Lemma 4. For any graph G with at least two vertices, tvs(G)≥2. Proof: Let G be graph with two vertices. Then degree of each vertex must be 1. With the integer 1, we can label only one vertex and the weight of that vertex will be 2 (both vertex and edge get the label 1). To label the other vertex, another integer is needed except 1. Therefore, at least two integers are required to label all the vertices of G.
Observe that, if it is possible to label the vertices and edges of a graph such that the weight of the vertices are consecutive integers starting from least possible weight, then tvs is achieved. Our aim is to label the vertices and edges of G in such a way that weight of the vertices are consecutive integers subject to the condition that the highest possible weight is least. Lemma 5. If |D k | = (k + 1)p + 1 (k > 1) for some positive integer p, then at least p + 1 integers are required to label the vertices of D k to follow vertex irregular total labeling. Proof: Using the integer 1, we can label only one vertex of D k . Each of the k edges incident on the vertex receive the label 1 and the vertex also gets label 1. Hence the weight of that vertex is k + 1.
From the above two lemmas it is observed that without increasing the value of tvs, we can label some extra vertices of higher degree. The number of extra vertices of D k+1 can be labeled by the integers used to label the vertices of D k is denoted by e k+1 . The values of e i , (i = 3, 4, . . ., ∆) can be determined by using previous two lemmas.

The Algorithm
The main idea of the proposed algorithm is follows. The algorithm proceeds by labeling the vertices in increasing order of degree of the given graph. At first, the sets D 1 , D 2 , . . . , D ∆ are computed. Depending on the cardinal number of these sets, tvs(G) is determined based on Lemma 4.
A formal description of the algorithm is given below.

Algorithm tvs
Input: A set of n(> 1) sorted intervals of a proper interval graph G. Output: Total vertex irregularity strength of G, that is, tvs(G).
Step 2: If | D 1 |= 0 then e 2 = 0, go to next step else if | D 1 |= 1 then T = T + 1, e 2 = 1 and go to next step else T = T + 2, e 2 = 3 and go to next step end if.

End tvs
The correctness of the algorithm follows from the previous lemmas. Theorem 2. For a proper interval graph G with n (> 1) vertices,

( ) ⌈ ⌉
Proof: Observe that, if G be a complete graph of order n then tvs(G) = 2 (Lemma 2). If G be not complete then tvs(G) > 2. Therefore, tvs(G) ≥ 2. Again, the tvs of a graph G will be maximum, if G consist of maximum number of vertices of less degree. Now, a connected proper interval graph with n vertices can have at most 2 leaf vertices and n−2 vertices of degree 2. In this case, G becomes a path P n of n vertices.
Therefore | D 1 |= 2 and | D 2 |= n− 2. The integers 1 and 2 are necessary to label the vertices of D 1 . Using these two integers, we can label 3 vertices of D 2 , that is, e 2 = 3. Hence, we have Note that, the algorithm tvs does not work for general interval graph as the Lemma 2 is not true for general interval graph. The maximum number of leaf vertex of an interval graph is n − 1. So, the algorithm needs modification depending on the number of leaf vertex to compute tvs of a general interval graph.

Proof:
The lower bound follows from the previous theorem.
Since maximum value of tvs can be attained when the graph has the maximum number of vertices of less degree. Therefore, an upper bound is attained if the graph has maximum number of leaf vertex. Now, a connected interval graph G with n vertices can have at most n − 1 vertices of degree 1 (leaf vertex). In that case, G becomes a star and ( ) ⌈ ⌉.

Conclusion
In this paper, an algorithm is designed to compute the exact value of tvs of proper interval graphs. Based on this algorithm the boundary values of tvs presented for proper interval graphs and interval graphs. Our approach can be generalized to determine the tvs of other intersection graphs, namely permutation graphs, trapezoid graphs and circular arc graphs. Future study can be done to find the value of tvs for these intersection graphs.