A set S of vertices of a graph G is called a decycling set if G-S is acyclic.The smallest size of a decycling set is called the decycling number of G and is denoted by ∇(G).In this paper,we investigate the decycling n...A set S of vertices of a graph G is called a decycling set if G-S is acyclic.The smallest size of a decycling set is called the decycling number of G and is denoted by ∇(G).In this paper,we investigate the decycling number of type-k Halin graphs,focusing on those that are formed from trees that have just two degrees k and 3.For any type-k Halin graph G of order n,we prove that(k-2)n+k^(2)-4k+5/(k-1)^(2)≤∇(G)≤n+k-3/k-1.The result not only supports the largest forest conjecture due to Albertson and Berman(1976),but also offers a tight lower bound for the decycling number of type-3 Halin graphs and several type-k Halin graphs.Moreover,a new formula to determine the cardinality of any decycling set S of a type-k Halin graph G is provided.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.11171114,11401576)Hotan Prefecture Science and Technology Bureau General Project(Grant No.20220212)。
文摘A set S of vertices of a graph G is called a decycling set if G-S is acyclic.The smallest size of a decycling set is called the decycling number of G and is denoted by ∇(G).In this paper,we investigate the decycling number of type-k Halin graphs,focusing on those that are formed from trees that have just two degrees k and 3.For any type-k Halin graph G of order n,we prove that(k-2)n+k^(2)-4k+5/(k-1)^(2)≤∇(G)≤n+k-3/k-1.The result not only supports the largest forest conjecture due to Albertson and Berman(1976),but also offers a tight lower bound for the decycling number of type-3 Halin graphs and several type-k Halin graphs.Moreover,a new formula to determine the cardinality of any decycling set S of a type-k Halin graph G is provided.
