Research Article Characterizations of Certain Types of Type 2 Soft Graphs Khizar Hayat , 1 Bing-Yuan Cao , 1,2 Muhammad Irfan Ali , 3 Faruk Karaaslan , 4 and Zejian Qin 1 1 School of Mathematics and Information Sciences, Guangzhou University, 510000 Guangzhou, China 2 Guangzhou Vocational College of Science and Technology, Guangzhou, Guangdong 510550, China 3 Department of Mathematics, Islamabad Model College for Girls, F-6/2, Islamabad, Pakistan 4 Department of Mathematics, Faculty of Sciences, C¸ankırı Karatekin University, 18100 C¸ankırı, Turkey Correspondence should be addressed to Bing-Yuan Cao; caobingy@163.com Received 4 March 2018; Accepted 29 August 2018; Published 20 September 2018 Academic Editor: Allan C. Peterson Copyright © 2018 Khizar Hayat et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e vertex-neighbors correspondence has an essential role in the structure of a graph. e type 2 soſt set is also based on the correspondence of initial parameters and underlying parameters. Recently, type 2 soſt graphs have been introduced. Structurally, it is a very efficient model of uncertainty to deal with graph neighbors and applicable in applied intelligence, computational analysis, and decision-making. e present paper characterizes type 2 soſt graphs on underlying subgraphs (regular subgraphs, irregular subgraphs, cycles, and trees) of a simple graph. We present regular type 2 soſt graphs, irregular type 2 soſt graphs, and type 2 soſt trees. Moreover, we introduce type 2 soſt cycles, type 2 soſt cut-nodes, and type 2 soſt bridges. Finally, we present some operations on type 2 soſt trees by presenting several examples to demonstrate these new concepts. 1. Preliminaries and Introduction A graph G = (, E) consists of a nonempty set of objects , called vertices, and a set E of two element subsets of  called edges. Two vertices  and  are adjacent if {, } ∈ E.A loop is an edge that connects a vertex to itself. A simple graph is an unweighted, undirected graph containing no multiple edges or graph loops. A graph G  = (  , E  ) is said to be a subgraph of G = (, E) if   ⊆ and E  ⊆ E. e graph neighborhoods of a vertex  in a graph is the set of all the vertices adjacent to  including  itself. e graph neighbors of a vertex  in a graph are the set of all the vertices adjacent to  excluding  itself. e eccentricity of the vertex  is the maximum distance from  to any vertex. e distance between two vertices  and  in a graph is the number of edges in a shortest path, denoted by (, ). e radius of a graph is the minimum eccentricity of any vertex . A graph G = (, E) is called a tree if it is connected and contains no cycles. Equivalently (and sometimes more useful), a tree is a connected graph on  vertices with exactly −1 edges. A forest is a disjoint union of trees. e degree of a vertex of a simple graph is the number of edges incident to the vertex. A regular graph is a graph where each vertex has the same number of neighbors. A graph that is not a regular graph is called irregular graph. A graph is called neighborly irregular graph if no two adjacent vertices have the same degree. A complete graph is a graph in which each pair of graph vertices is connected by an edge. For basic definitions of graphs see [1–3]. Soſt set theory [4], firstly initiated by Molodtsov, is a new mathematical tool for dealing with uncertainties. Some fruitful operations, soſt set theory, are presented by Maji et al. [5] and Ali et al. [6]. We refer to Molodtsov’s soſt sets as type 1 soſt sets (briefly 1). Let  be a set of parameters that can have an arbitrary nature (numbers, functions, sets of words, etc.). Let  be a universe and the power set of  is denoted by (). e soſt set is defined as follows. Definition 1 (see [5]). A pair (, ) is called a soſt set over , where  is a mapping given by  : → (). Note that the set of all 1 over  will be denoted by (). Many researchers take attention at applicability of soſt sets in real and practical problems. In recent years, research Hindawi Discrete Dynamics in Nature and Society Volume 2018, Article ID 8535703, 15 pages https://doi.org/10.1155/2018/8535703