Independent Sets in Classes Related to Chair/Fork-free Graphs

Coloring graph classes with no induced fork via perfect divisibility

For a graph $G$, $χ(G)$ will denote its chromatic number, and $ω(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. An integer-valued function $f$ is called a $χ$-binding function for a hereditary class of graphs $\cal C$ if $χ(G) \leq f(ω(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $χ$-binding function for t…

Reed's conjecture on some special classes of graphs

Reed conjectured that for any graph $G$, $χ(G) \leq \lceil \frac{ω(G)+Δ(G)+1}{2}\rceil$, where $χ(G)$, $ω(G)$, and $Δ(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we verify this conjecture for some special classes of graphs, in particular for subclasses of $P_5$-free graphs or $Chair$-free graphs.

Polynomial Cases for the Vertex Coloring Problem

The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.

Partitioning H-Free Graphs of Bounded Diameter

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by …

Bounding the Clique-Width of $H$-free Chordal Graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex $P_4$-extensions $H$ for which the class of $H$-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of $H$-free chordal graphs of bounde…

Flood-It 与珠宝——Cograph 推广的游戏复杂度刻画

Flood-It with Jewelry -- Characterizing the Game Complexity for Cograph Generalizations

本文研究 Flood-It 游戏在禁止某些诱导子图的图类上的复杂度，给出了 Free Flood-It 在禁止 bull、gem 和 P5 的图上的多项式时间算法，并证明了两种变体在 thin-spider 图上的 NP-完全性，确定了 1024 个图类中 896 个的复杂度。