For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$. We are interested in the complexity of the problem $H$-{\sc Coloring}, which asks for the existence of an $H$-coloring of an input graph $G$. In particular, we consider $H$-{\sc Coloring} of $F$-free graphs, where $F$ is a fixed graph and $H$ is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-{\sc Coloring} of $P_t$-free graphs.
We show that for every odd $k \geq 5$ the $C_k$-{\sc Coloring} problem, even in the list variant, can be solved in polynomial time in $P_9$-free graphs. The algorithm extends for the case of list version of $C_k$-{\sc Coloring}, where $k$ is an even number of length at least 10.
On the other hand, we prove that if some component of $F$ is not a subgraph of a subdividecd claw, then the following problems are NP-complete in $F$-free graphs: a)extension version of $C_k$-{\sc Coloring} for every odd $k \geq 5$, b) list version of $C_k$-{\sc Coloring} for every even $k \geq 6$.
For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the $H$-Coloring problem the graph $H$ is fixed and we ask whether an instance graph $G$ admits an $H$-coloring. A generalization of this problem is $H$-ColoringExt, where some vertices of $G$ are already mapped to vertices of $H$ and we ask if this partial mapping can be extended to an $H$-coloring. We study the complexity of variants of $H$-Coloring in $F$-free graphs, i.e., graphs excluding a fixed graph $F$ as an induced subgraph. For integers $a,b,c \geq 1$, by $S_{a,b,c}$ we denote the …
We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph $G$, a non-trivial hereditary property $Π$ and an integer parameter $k$, the general problem $P(G,Π,k)$ asks whether there exists $k$ vertices of $G$ that induce a subgraph satisfying property $Π$. This problem, $P(G,Π,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[1]-complete by Khot and Raman, if $Π$ includes all trivial graphs but not all complete graphs and vic…
A large number of NP-hard graph problems become polynomial-time solvable on graph classes where the mim-width is bounded and quickly computable. Hence, when solving such problems on special graph classes, it is helpful to know whether the graph class under consideration has bounded mim-width. We first extend the toolkit for proving (un)boundedness of mim-width of graph classes. This enables us to initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes. For a given graph $H$, the class of $H$-free graphs has bounded mim-width if and only if it has bo…
In this paper, we have obtained the total chromatic as well as equitable and neighborhood sum distinguishing total chromatic numbers of some classes of the circulant graphs.
A k-role coloring of a graph G is an assignment of k colors to the vertices of G such that if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n-role coloring. While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k-role coloring is a notoriously hard problem for k greater than 1. In fact, it is known that k-Role coloring is NP-complete for k greater than 1 on arbitrary graphs. There has been extensive research …
We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to $(H_1,H_2)$-free graphs and a dichotomy for the latter problem restricted to $H$-free graphs. We find that there exists an infinite family of graphs $H$ such that Vertex Steiner Tree is polynomial-time solvable for $H$-free graphs, whereas there exist only two graphs $H$ for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree…
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first as…
For a graph class ${\cal H}$, the graph parameters elimination distance to ${\cal H}$ (denoted by ${\bf ed}_{\cal H}$) [Bulian and Dawar, Algorithmica, 2016], and ${\cal H}$-treewidth (denoted by ${\bf tw}_{\cal H}$) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class ${\cal H}$. Here, the torso of a vertex set $S$ in a graph $G$ is the graph with vertex set $S$ and an edge between two vertices $u, v \in S$ if there is a path between $u$ and $v$ in $G$ whose internal vertices all lie outside …
Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which generalize the well-known graph invariant degeneracy (also called k-core number). Being NP-hard, weak-coloring numbers were previously computed on real-world graphs mainly via incremental heuristics. We study whether it is feasible to augment such heuristics with exponential-time subprocedures that kick in when a desired upper bound on the weak coloring number i…
