On the Connectivity of Token Graphs of Trees

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal to the minimum degree of $F_k(G)$

On the spectra of token graphs of cycles and other graphs

The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs $O_r$ for all $r$, and the multipartite complete graphs $K_{n_1,n_2,\ldots,n_r}$ for all $n_1,n_2,\ldots,n_r$ In the case of cycles, we present a new method that allows us to compute the whole spectrum of $F_2(C_n)$. This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of $F_2(\textit{}C_n)$

On the spectra of token graphs of cycles and other graphs

© 2023 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The k-token graph of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs for all r, and the multipartite complete graphs for all In the case of cycles, we present a new method that allows us to compute the whole spectrum of . This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of .Peer ReviewedPostprint (author's final draft

Connectivity properties of some transformation graphs

Many combinatorial problems can be formulated as \can we transform configuration 1 into configuration 2 if certain transformations are allowed?" In order to study such questions, we introduce a so-called transformation graph. This graph has the set of all possible configurations as its vertex set, and there is an edge between two configurations if one configuration can be obtained from the other by one of the allowed transformations. Then a question like \can we go from one configuration to another one" becomes a question about connectivity properties of transformation graphs. In this thesis, we study the following types of transformation graphs in particular: Labelled Token Graphs: Here configurations are arrangements of labelled tokens on a given graph, and we can go from one arrangement to another one by moving one token at a time along an edge of the given graph. We classify all cases when labelled token graphs are connected, and classify all pairs of arrangements that are in the same component. We also look at the problem how hard it is to determine the length of the shortest path between two arrangements. Strong k-Colour Graphs: For this transformation graph, the configurations are the proper vertex-colourings of a given graph with k colours, in which all k colours are actually used. We call such a colouring a strong k-colouring. We study the problem when we can transform any strong k-colouring into any other one by recolouring one vertex at a time, always maintaining a strong k-colouring. For certain classes of graphs, we can completely determine when the transformation graph of strong k-colourings is connected

Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs

We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of $k$.Comment: 20 pages, 1 figure, 28 reference

The Complexity of Change

Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two solutions of a SAT-instance, can I transform the first solution into the second one, by changing the truth value one variable at a time, and always maintaining a solution of the SAT-instance? Other examples can be found in many classical puzzles, such as the 15-Puzzle and Rubik's Cube. In this survey we shall give an overview of some older and more recent work on this type of problem. The emphasis will be on the computational complexity of the problems: how hard is it to decide if a certain transformation is possible or not?Comment: 28 pages, 6 figure