A Polynomial Algorithm for Weighted Toughness of Interval Graphs

The concept of toughness, introduced by Chv a � tal, has been widely used as an important invulnerability parameter. This parameter is generalized to weighted graphs, and the concept of weighted toughness is proposed. A polynomial algorithm for computing the weighted toughness of interval graphs is


Introduction
e concept of invulnerability was proposed in the early research on the connectivity of communication networks, which reflect the ability of a network to resist deliberate damage from the outside [1]. As a widely used invulnerability parameter, toughness was introduced by Chva �tal in 1973.
Definition 1 (see [2]). Let G be a noncomplete graph. e toughness of G is defined as If there exists X * ⊆V(G) such that t(G) � |X * |/ω(G − X * ), then X * is called a t-tough set, denoted as t-set. In particular, for the complete graph K n , define t(K n ) � ∞.
In reality, the vertices of a graph have different roles. Usually, vertex-weighted graphs are used to represent such network models, that is, each vertex is associated with a real number to distinguish such a difference. To measure the invulnerability of networks, we introduce the concept of weighted toughness of graphs.

Definition 2.
Let G be a noncomplete vertex-weighted graph. e weighted toughness of G is defined as where w: V ⟶ R + is a nonnegative weight function and w(X) � v∈X w(v) is the weight of the vertex cut X. For a vertex-weighted complete graph K n , define t w (K n ) � ∞. By the definition, the greater the weighted toughness, the stronger the invulnerability of the vertex-weighted graph.
When the weights of all vertices are equal, weighted toughness is equivalent to toughness. e problem of toughness computation is NP-hard [3]. However, there exists a polynomial algorithm for toughness computation of interval graphs [4]. e definition of the interval graph is given below.
Definition 4 (see [5]). A graph G is called an interval graph, if ∀v ∈ V(G) corresponds to a closed interval I v � [a v , b v ], and uv ∈ E(G) if and only if I u ∩ I v ≠ ∅. We call I v v∈V the interval representation of G.
Due to the special structure and properties, interval graphs are widely used in theoretical research and engineering practice. For example, the archeology chronological order of unearthed items [6], the scheduling problems in computer science [7], and the determination of biology model of DNA sequences [8]. Many scholars have studied the algorithm and structure of interval graphs. Kratsch et al. gave a polynomial time algorithm for computing scattering numbers and toughness of interval graphs [4]. Broersma et al. gave a linear time algorithm for computing scattering numbers of interval graphs [9]. Li et al. gave a polynomial algorithm for computing weighted scattering number of interval graphs [10].
In this paper, we consider the algorithm of computing weighted toughness of interval graphs. e following work is arranged in two parts. In Section 2, some elementary definitions and notations, as well as some preliminary results are given. An algorithm for computing the weighted toughness of interval graphs is given based on the investigation of properties of achieving cuts and local minimum cuts. In Section 3, we summarize our work by the complexity analysis of the algorithm.
is paper only considers the finite simple undirected graphs. For the terminology and notations not defined here, we refer the reader to [11].

An Algorithm for Computing Weighted Toughness of Interval Graphs
In this section, we firstly investigate the properties of achieving cuts and local minimum cuts of interval graphs. e number of vertex cut of a graph increases with its order exponentially. Fortunately, it is not necessary to consider all vertex cuts when computing the weighted toughness of interval graphs.
Definition 5. (see [12]). Let G be an interval graph and X be a vertex cut. If for every proper subset Y ⊂ X and ω(G − Y) < ω(G − X), then X is called a strong cut of G. Theorem 1. Let G be an interval graph with nonnegative weights' function w: V ⟶ R + . en, any achieving cut of G is a strong cut.
Proof. Let X be an achieving cut of G but not a strong cut. By Definition 5, there must exist a vertex u ∈ X such that is contradicts to that X is an achieving cut of G. e proof is completed. □ Definition 6 (see [12]). Let G be an interval graph. Consider a point x such that min i b i < x < max i a i . If the end point immediately to the left of x is right end point and the end point immediately to the right of x is a left end point, then C(x) is called a minimal local cut of G. Lemma 1 (see [12]). Any strong cut of an interval graph can be expressed as a union of minimal local cuts.
By the definition of interval representation, a < a i+1 < a i+2 . Since eorem 3 shows that, for an interval graph, nonadjacent minimal local are disjoint. e algorithm for finding the union of minimal local cuts with nonempty intersection is given below.
Step 5 finds union of two minimal local cuts with nonempty intersection, which need |C(α i )||C(α i+1 )| comparisons. Similarly, Step 7 needs |X r− 1 ||S j | comparisons. It is easy to know that Step 2 to Step 8 need k − 1 circulations, and the total computations does not exceed Since an interval graph of order n has at most n − 2 minimal local cuts and each minimal local cut contains at most n − 2 vertices, the complexity of Algorithm 2 is O(n 3 ). e proof is completed. By the discussion above, to computing the weighted toughness of an interval graph, it is sufficient to consider the union of minimally local cuts with nonempty intersection, as well as the number of connected components of the remaining subgraphs. Proof. e total computations of Step 2 is n. In Step 3, obtaining graph G � G − S 0 requires 2|V||S 0 | operations.
is example shows that the value of weighted toughness is not only related to the structure and weights of the graph but also related to the way of weight association.

Remark 2.
e weight of G 1 and G 2 are randomly generated.

Conclusion Remarks
At last, we consider the complexity of Algorithm 3. Let A(G) � (a ij ) n×n be the adjacency matrix of an interval graph G of order n. In Algorithm 3, computing w(X) and  Figure 1: An interval graph G and its interval representation.