For a positive integer k , a {k}-Roman dominating function of a graph G = (V,E) is a function f: V —> {0,1,... ,k} satisfying f(N(v)) ≥ k for each vertex v ε V with f(v) = 0. Every graph G satisfes γ { Rk } (G) ≤ kγ(G) , where γ { Rk } ( G ) denotes the minimum weight of a { k }-Roman dominating function of G and γ(G) is the domination number of G . In this work we study graphs for which the equality is reached, called {k}-Roman graphs. This extends the concept of { k }-Roman trees studied by Wang et al. in 2021 to general graphs. We prove that for every k ≥ 3, the problem of recognizing { k }-Roman graphs is NP-hard, even when restricted to split graphs. We provide partial answers to the question of which split graphs are {2}-Roman: we characterize {2}-Roman split graphs that can be decomposed with respect to the split join operation into two smaller split graphs and classify the { k }-Roman property within two specific families of split graphs that are prime with respect to the split join operation: suns and their complements.

For a tree decomposition $\mathcal{T}$ of a graph $G$, by $μ(\mathcal{T})$ we denote the size of a largest induced matching in $G$ all of whose edges intersect one bag of $\mathcal{T}$. Induced matching treewidth of a graph $G$ is the minimum value of $μ(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Yolov [SODA 2018] proved that Max Weight Independent Set can be solved in polynomial time for graphs of bounded induced matching treewidth. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO$_2$ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth.

Twin-width is a width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler genus $g$ is $18\sqrt{47g}+O(1)$, which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus $g$ that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size $\max\{8,32g-27\}$.

Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $({tw},\omega)$-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and $k$-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs $H$ for which the class of graphs excluding $H$ is $({tw},\omega)$-bounded. Our results yield an infinite family of $\chi$-bounded induced-minor-closed graph classes and imply that the class of 1-perfectly orientable graphs is $({tw},\omega)$-bounded, leading to linear-time algorithms for $k$-coloring 1-perfectly orientable graphs for every fixed $k$. This answers a question of Brešar, Hartinger, Kos, and Milanič from 2018 and one of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of $({tw},\omega)$-boundedness related to list $k$-coloring and clique problems. In addition, we propose a question about the complexity of the maximum weight independent set problem in $({tw},\omega)$-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor.