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Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is $\#\textsf{P}$-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length $2$ or paths of length $1$. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling., Comment: Full version of paper to appear in SWAT 2024; arXiv admin note: text overlap with arXiv:2305.01851

We study the problem of reconfiguring one minimum $s$-$t$-separator $A$ into another minimum $s$-$t$-separator $B$ in some $n$-vertex graph $G$ containing two non-adjacent vertices $s$ and $t$. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming $A$ into $B$. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most $\ell$ jumps can transform $A$ into $B$ is an $\textsf{NP}$-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size $k$ of the minimum \stseps and when parameterized by the number of jumps $\ell$. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complete the picture by designing a kernel with $\mathcal{O}(\ell^2)$ vertices and edges for the length $\ell$ of the sequence as a parameter., Comment: 37 pages, 9 figures

In this paper, we examine the computational complexity of enumeration in certain genome rearrangement models. We first show that the Pairwise Rearrangement problem in the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010) is $\#\textsf{P}$-complete under polynomial-time Turing reductions. Next, we show that in the Single Cut or Join model (Feijao & Meidanis, IEEE ACM Trans. Comp. Biol. Bioinf. 2011), the problem of enumerating all medians ($\#$Median) is logspace-computable ($\textsf{FL}$), improving upon the previous polynomial-time ($\textsf{FP}$) bound of Mikl\'os & Smith (RECOMB 2015)., Comment: Full version of paper that appeared in COCOON 2023: https://doi.org/10.1007/978-3-031-49190-0_1

A \emph{Kempe chain} on colors $a$ and $b$ is a component of the subgraph induced by colors $a$ and $b$. A \emph{Kempe change} is the operation of interchanging the colors of some Kempe chain. For a list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe change is \emph{$L$-valid} for $\varphi$ if performing the Kempe change yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from the other by a sequence of $L$-valid Kempe changes. A \emph{degree-assignment} is a list-assignment $L$ such that $L(v)\ge d(v)$ for every $v\in V(G)$. Cranston and Mahmoud (\emph{Combinatorica}, 2023) asked: For which graphs $G$ and degree-assignment $L$ of $G$ is it true that all the $L$-colorings of $G$ are $L$-equivalent? We prove that for every 4-connected graph $G$ which is not complete and every degree-assignment $L$ of $G$, all $L$-colorings of $G$ are $L$-equivalent., Comment: 12 pages; major changes in presentation

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs $\{G-v: v\in V(G)\}$. Let $diam(G)$ and $\kappa(G)$ denote the diameter and the connectivity of a graph $G$, respectively, and let $\mathcal{G}_2:=\{G: \textrm{diam}(G)=2\}$ and $\mathcal{G}_3:=\{G:\textrm{diam}(G)=\textrm{diam}(\overline{G})=3\}$. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in $\mathcal{G}_2\cup \mathcal{G}_3$. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_2\cup \mathcal{G}_3$ with $\kappa(G)=2$. Moreover, they asked whether the result still holds if $\kappa(G)\ge 3$. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in $\mathcal{G}_2\cup\mathcal{G}_3$ which contain triangles.) In this paper, we give a partial solution to their question by showing that the Reconstruction Conjecture holds for every triangle-free graph $G$ in $\mathcal{G}_3$ and every triangle-free graph $G$ in $\mathcal{G}_2$ with $\kappa(G)=3$. We also prove similar results about the Edge Reconstruction Conjecture., Comment: 11 pages, 3 figures

The coloring reconfiguration graph $\mathcal{C}_k(G)$ has as its vertex set all the proper $k$-colorings of $G$, and two vertices in $\mathcal{C}_k(G)$ are adjacent if their corresponding $k$-colorings differ on a single vertex. Cereceda conjectured that if an $n$-vertex graph $G$ is $d$-degenerate and $k\geq d+2$, then the diameter of $\mathcal{C}_k(G)$ is $O(n^2)$. Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$ for every planar graph $G$ with no 3-cycles and no 5-cycles., Comment: 7 pages, 3 figures, corrects a few errors in the previous version

An $\alpha,\beta$-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors $\alpha$ and $\beta$. Two $k$-colorings of a graph are $k$-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than $k$ colors). Las Vergnas and Meyniel showed that if a graph is $(k-1)$-degenerate, then each pair of its $k$-colorings are $k$-Kempe equivalent. Mohar conjectured the same conclusion for connected $k$-regular graphs. This was proved for $k=3$ by Feghali, Johnson, and Paulusma (with a single exception $K_2\square K_3$, also called the 3-prism) and for $k\ge 4$ by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap is called $L$-valid for $\varphi$ if performing the Kempe swap yields another $L$-coloring. Two $L$-colorings are called $L$-equivalent if we can form one from the other by a sequence of $L$-valid Kempe swaps. Let $G$ be a connected $k$-regular graph with $k\ge 3$. We prove that if $L$ is a $k$-assignment, then all $L$-colorings are $L$-equivalent (again with a single exception $K_2 \square K_3$). When $k\ge 4$, the proof is completely self-contained, so implies an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let $H$ be a graph such that for every degree-assignment $L_H$ all $L_H$-colorings are $L_H$-equivalent. If $G$ is a connected graph that contains $H$ as an induced subgraph, then for every degree-assignment $L_G$ for $G$ all $L_G$-colorings are $L_G$-equivalent., Comment: 29 pages, 12 figures; second version extends the main result to cliques, which were previously excluded; third version incorporates reviewer feedback; to appear in Combinatorica

Enabling empathetic behavior in Arabic dialogue agents is an important aspect of building human-like conversational models. While Arabic Natural Language Processing has seen significant advances in Natural Language Understanding (NLU) with language models such as AraBERT, Natural Language Generation (NLG) remains a challenge. The shortcomings of NLG encoder-decoder models are primarily due to the lack of Arabic datasets suitable to train NLG models such as conversational agents. To overcome this issue, we propose a transformer-based encoder-decoder initialized with AraBERT parameters. By initializing the weights of the encoder and decoder with AraBERT pre-trained weights, our model was able to leverage knowledge transfer and boost performance in response generation. To enable empathy in our conversational model, we train it using the ArabicEmpatheticDialogues dataset and achieve high performance in empathetic response generation. Specifically, our model achieved a low perplexity value of 17.0 and an increase in 5 BLEU points compared to the previous state-of-the-art model. Also, our proposed model was rated highly by 85 human evaluators, validating its high capability in exhibiting empathy while generating relevant and fluent responses in open-domain settings.

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the same is true for 5-colorings. We answer their question affirmatively when $m,n\ge 6$. Further, we show that if $G$ is 6-regular with a toroidal embedding where every non-contractible cycle has length at least 7, then all 5-colorings of $G$ are 5-equivalent. Our results relate to the antiferromagnetic Pott's model in statistical mechanics., Comment: 17 pages, 16 figures; 4th version incorporates minor referee feedback; to appear in European Journal of Combinatorics