On chordal and perfect plane triangulations Sameera M Salam, Nandini J Warrier, Daphna Chacko, K. Murali Krishnan, Sudeep K. S. Department of Computer Science and Engineering, National Institute of Technology Calicut, Kerala, India 673601 February 20, 2020 Abstract It is shown that a plane near triangulated graph is chordal if and only if it does not contain any induced wheel W n for n ≥ 5. A necessary and sufficient condition for a W 5 free plane near triangulation to be perfect is also derived. Keywords: Plane triangulated graphs, plane near triangulated graphs, Chordal graphs, Perfect graphs 1. Introduction A plane embedding of a (planar) graph G is called a triangulation if the boundary of every face is a cycle of length three. A plane embedding of a simple maximal planar graph is a plane triangulation [1]. If the above definition is relaxed to permit the boundary of the outer face to be a cycle of length exceeding three, the embedding is called a plane near triangulation. A graph G is said to contain a chordless cycle if it has an induced cycle on four or more vertices. A graph that does not contain any chordless cycle is called a chordal graph [2]. G is said to be perfect if neither G nor its complement contain any chordless odd cycles. Structural properties of maximal planar graphs and some of their subfamilies like Appollo- nian Networks have been investigated in the literature [3] [4][5][6]. Here we investigate structural characterizations for chordal and perfect plane near triangulations. The results hold true for plane triangulations as well. In Section 3, we derive a simple characterization for chordal plane near triangulations that depends solely on the structure of the local neighbourhood of individual vertices. Though struc- tural characterization for perfect plane triangulations has been investigated in [7], to the best of our knowledge, no characterization that depends only on the local neighbourhoods of individual vertices is known. In Section 4 we investigate a relatively simple structural characterization for perfect plane near triangulations. Though a local characterization for perfect plane triangulations (or plane near triangulations) is unlikely, plane near triangulations that do not contain any induced W 5 (See Definition 2.1) admits a local characterization. A proof for the characterization is given in Section 5. The following section establishes some notation and definitions. Email addresses: shemi.nazir@gmail.com (Sameera M Salam), nandini.wj@gmail.com (Nandini J Warrier), daphna.chacko@gmail.com (Daphna Chacko), kmurali@nitc.ac.in (K. Murali Krishnan), sudeep@nitc.ac.in (Sudeep K. S.) 1 arXiv:1701.03447v2 [math.CO] 13 Sep 2017