Intersection of longest paths in graph classes
نویسندگان
چکیده
منابع مشابه
Intersection of Longest Paths in a Graph
In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected g...
متن کاملOn Computing Longest Paths in Small Graph Classes
The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that t...
متن کاملNonempty intersection of longest paths in series-parallel graphs
In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai’s question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain ...
متن کاملA note on longest paths in circular arc graph
As observed by Rautenbach and Sereni [SIAM J. Discrete Math. 28 (2014) 335–341] there is a gap in the proof of the theorem of Balister et al. [Combin. Probab. Comput. 13 (2004) 311–317], which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.
متن کاملTransversals of Longest Paths
Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and tw(G) be the treewidth of G. We prove that lpt(G) ≤ max{1,ω(G)− 2} when G is a connected chordal graph; that lpt(G) = 1 when G is a connected bipartite permutation graph or a connected full substar graph; and that lpt(G) ≤ tw(G) for any co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete Applied Mathematics
سال: 2020
ISSN: 0166-218X
DOI: 10.1016/j.dam.2019.03.022