Abstract:In this paper, we study the class of graphs
scriptC defined by excluding the following structures as induced subgraphs: theta, pyramid, 1‐wheel, and 3‐wheel. We describe the structure of graphs in
scriptC, and we give a polynomial‐time recognition algorithm for this class. We also prove that
K
4‐free graphs in
scriptC are 4‐colorable. We remark that
scriptC includes the class of chordal graphs, as well as the class of line graphs of triangle‐free graphs.
“…Chudnovsky and Seymour are interested in thetas because they trivially imply an even hole. They developed the previously only known polynomial-time algorithm, running in O(n 11 ) time, for detecting thetas in G via solving the three-in-a-tree problem on O(n 7 ) subgraphs of G. Thus, Theorem 1.1 reduces the time to Õ(n 9 ). Moreover, we show in Lemma 6.1 that thetas in G can be detected via solving the three-in-a-tree problem on O(mn 2 ) n-vertex graphs, leading to an Õ(n 6 )-time algorithm as stated in Theorem 1.2.…”
Section: Discussionmentioning
confidence: 99%
“…See [1,4,11,14,20,21,22,23,35,44,45,47,49,51,55,70] for more related work on graph detection, recognition, and characterization. Also see [12, Appendix A] for a survey of the recognition complexity of more than 160 graph classes.…”
Section: Other Related Workmentioning
confidence: 99%
“…Chang et al's algorithm consists of two O(n11 )-time phases. The first phase detects beetles in O(n11 ) time, which is now reduced to O(n 7 ) time by Theorem 1.5.…”
mentioning
confidence: 99%
“…Chang et al's algorithm consists of two O(n11 )-time phases. The first phase detects beetles in O(n11 ) time, which is now reduced to O(n 7 ) time by Theorem 1.5. The second phase maintains a set T of induced subgraphs of G with the property that if G is even-hole-free, then so is each graph in T until either T becomes empty or an H ∈ T is found to contain even holes.…”
The three-in-a-tree problem is to determine if a simple undirected graph contains an induced subgraph which is a tree connecting three given vertices. Based on a beautiful characterization that is proved in more than twenty pages, Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known polynomial-time algorithm, running in O(mn 2 ) time, to solve the three-in-a-tree problem on an n-vertex m-edge graph. Their three-in-a-tree algorithm has become a critical subroutine in several state-of-the-art graph recognition and detection algorithms.In this paper we solve the three-in-a-tree problem in Õ(m) time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes. Our result is based on a new and more constructive characterization than that of Chudnovsky and Seymour. Our new characterization is stronger than the original, and our proof implies a new simpler proof for the original characterization. The improved characterization gains the first factor n in speed. The remaining improvement is based on dynamic graph algorithms.
“…Chudnovsky and Seymour are interested in thetas because they trivially imply an even hole. They developed the previously only known polynomial-time algorithm, running in O(n 11 ) time, for detecting thetas in G via solving the three-in-a-tree problem on O(n 7 ) subgraphs of G. Thus, Theorem 1.1 reduces the time to Õ(n 9 ). Moreover, we show in Lemma 6.1 that thetas in G can be detected via solving the three-in-a-tree problem on O(mn 2 ) n-vertex graphs, leading to an Õ(n 6 )-time algorithm as stated in Theorem 1.2.…”
Section: Discussionmentioning
confidence: 99%
“…See [1,4,11,14,20,21,22,23,35,44,45,47,49,51,55,70] for more related work on graph detection, recognition, and characterization. Also see [12, Appendix A] for a survey of the recognition complexity of more than 160 graph classes.…”
Section: Other Related Workmentioning
confidence: 99%
“…Chang et al's algorithm consists of two O(n11 )-time phases. The first phase detects beetles in O(n11 ) time, which is now reduced to O(n 7 ) time by Theorem 1.5.…”
mentioning
confidence: 99%
“…Chang et al's algorithm consists of two O(n11 )-time phases. The first phase detects beetles in O(n11 ) time, which is now reduced to O(n 7 ) time by Theorem 1.5. The second phase maintains a set T of induced subgraphs of G with the property that if G is even-hole-free, then so is each graph in T until either T becomes empty or an H ∈ T is found to contain even holes.…”
The three-in-a-tree problem is to determine if a simple undirected graph contains an induced subgraph which is a tree connecting three given vertices. Based on a beautiful characterization that is proved in more than twenty pages, Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known polynomial-time algorithm, running in O(mn 2 ) time, to solve the three-in-a-tree problem on an n-vertex m-edge graph. Their three-in-a-tree algorithm has become a critical subroutine in several state-of-the-art graph recognition and detection algorithms.In this paper we solve the three-in-a-tree problem in Õ(m) time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes. Our result is based on a new and more constructive characterization than that of Chudnovsky and Seymour. Our new characterization is stronger than the original, and our proof implies a new simpler proof for the original characterization. The improved characterization gains the first factor n in speed. The remaining improvement is based on dynamic graph algorithms.
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