Schedule for: 17w5066 - Combinatorial Reconfiguration

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $I$ be an independent set of a graph $G$. Imagine that a token is located on any vertex of $I$. We can now move the tokens of $I$ along the edges of the graph as long as the set of tokens still defines an independent set of $G$. Given two independent sets $I$ and $J$, the Token Sliding problem consists in deciding whether there exists a sequence of independent sets which transforms $I$ into $J$ so that every pair of consecutive independent sets of the sequence can be obtained via a token move. This problem is known to be PSPACE-complete even on planar graphs. Demaine et al. [ISAAC'14] asked whether the Token Sliding reconfiguration problem is polynomial time solvable on interval graphs and more generally in chordal graphs. Yamada and Uehara [WALCOM'16] showed that a polynomial time transformation can be found in proper interval graphs. We answer the first question of Demaine et al. and generalize the result of Yamada and Uehara by showing that we can decide in polynomial time whether two independent sets of an interval graph are in the same connected component. Moreover, we answer similar questions by showing that: (i) determining if there exists a token sliding transformation between every pair of $k$-independent sets in an interval graph can be decided in polynomial time; (ii) deciding this problem becomes co-NP-hard and even co-W[2]-hard (parameterized by the size of the independent set) on split graphs, a sub-class of chordal graphs.

Two proper $k$-colorings of $G$ are incident if they differ on exactly one color. We can define the $k$-reconfiguration graph of $G$ where the vertices are the proper $k$-colorings of $G$ and we create an edge between two colorings if they are incident. One of the main conjectures in the reconfiguration community is the one by Cereceda, stating the following: for any $k$-degenerate graph $G$, the diameter of the $(k+2)$-reconfiguration graph of $G$ is quadratic in $|G|$. We will discuss partial results, generalization and open problems related to this conjecture.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.