An Improved Algorithm for Finding Maximum Outerplanar Subgraphs

We study the NP-complete Maximum Outerplanar Subgraph problem. The previous best known approximation ratio for this problem is 2/3. We propose a new approximation algorithm which improves the ratio to 7/10

An Algebraic Approach to the Non-chromatic Adherence of the DP Color Function

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. A function $f$ is chromatic-adherent if for every graph $G$, $f(G,a) = P(G,a)$ for some $a \geq \chi(G)$ implies that $f(G,m) = P(G,m)$ for all $m \geq a$. It is known that the DP color function is not chromatic-adherent, but there are only two known graphs that demonstrate this. Suppose $G$ is an $n$-vertex graph and $\mathcal{H}$ is a 3-fold cover of $G$, in this paper we associate with $\mathcal{H}$ a polynomial $f_{G, \mathcal{H}} \in \mathbb{F}_3[x_1, \ldots, x_n]$ so that the number of non-zeros of $f_{G, \mathcal{H}}$ equals the number of $\mathcal{H}$-colorings of $G$. We then use a well-known result of Alon and F\"{u}redi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on $P_{DP}(G,3)$ when $2n > |E(G)|$. Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP color function.Comment: 8 pages. arXiv admin note: text overlap with arXiv:2107.08154, arXiv:2110.0405

Reductions for the Stable Set Problem

One approach to finding a maximum stable set (MSS) in a graph is to try to reduce the size of the problem by transforming the problem into an equivalent problem on a smaller graph. This paper introduces several new reductions for the MSS problem, extends several well-known reductions to the maximum weight stable set (MWSS) problem, demonstrates how reductions for the generalized stable set problem can be used in conjunction with probing to produce powerful new reductions for both the MSS and MWSS problems, and shows how hypergraphs can be used to expand the capabilities of clique projections. The effectiveness of these new reduction techniques are illustrated on the DIMACS benchmark graphs, planar graphs, and a set of challenging MSS problems arising from Steiner Triple Systems