We present a logspace algorithm for computing a canonical interval representation and a canonical labeling of interval graphs. As a consequence, the isomorphism and automorphism problems for interval graphs are solvable in logspace.

We prove a complexity classification theorem that divides the six-vertex model on graphs into exactly three types. For every setting of the parameters of the model, the computation of the partition function is precisely: Either (1) solvable in polynomial time for every graph, or (2) #P-hard for general graphs but solvable in polynomial time for planar graphs, or (3) #P-hard even for planar grap...

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diamete...

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For general m-edge and n-vertex graphs, it is well-known to be solvable in O(m √ n) time. We develop the first linear-time algorithm to find maximumcardinality matchings on cocomparability graphs, a prominent subclass of perfect graphs that contains interval graphs as well as permutati...

The game of peg solitaire on graphs was introduced by Beeler and Hoilman in 2011. In this game, pegs are initially placed on all but one vertex of a graph GG. If xyz forms a path in GG and there are pegs on vertices x and y but not z, then a jump places a peg on z and removes the pegs from x and y. A graph is called solvable if, for some configuration of pegs occupying all but one vertex, some ...

For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. H-graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This class was introduced in the early 1990s by B́ıró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fu...

Consider an undirected graph modeling a social network, where the vertices represent users, and the edges do connections among them. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her opinion, and then it spreads along the edges in the graphs. The objective of every player is to maximize the number of vertices the opinion infects. In t...