D. Adam Lazzarato

Bio: D. Adam Lazzarato is an academic researcher from Wilfrid Laurier University. The author has contributed to research in topics: 1-planar graph & Chordal graph. The author has an hindex of 2, co-authored 2 publications receiving 40 citations.

Polynomial-time algorithms for minimum weighted colorings of ($P_5, \bar{P}_5$)-free graphs and related graph classes.

Abstract: We design an $O(n^3)$ algorithm to find a minimum weighted coloring of a ($P_5, \bar{P}_5$)-free graph. Furthermore, the same technique can be used to solve the same problem for several classes of graphs, defined by forbidden induced subgraphs, such as (diamond, co-diamond)-free graphs.

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Abstract: For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.

A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs

Abstract: For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u) eq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.

Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Abstract: If a graph has no induced subgraph isomorphic to any graph in a finite family $$\{H_1,\ldots ,H_p\}$$, it is said to be $$H_1,\ldots ,H_p$$-free. The class of $$H$$-free graphs has bounded clique-width if and only if $$H$$ is an induced subgraph of the 4-vertex path $$P_4$$. We study the unboundedness of the clique-width of graph classes defined by two forbidden induced subgraphs $$H_1$$ and $$H_2$$. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of $$H_1,H_2$$-free graphsifor all pairs $$H_1,H_2$$, both of which are connected, except two non-equivalent cases, andiifor all pairs $$H_1,H_2$$, at least one of which is not connected, excepti?ź11 non-equivalent cases. We also consider classes characterized by forbidding a finite family of graphs $$\{H_1,\ldots ,H_p\}$$ as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colouring problem restricted to $$H_1,H_2$$-free graphs.

Clique-width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Abstract: If a graph has no induced subgraph isomorphic to any graph in a finite family $\{H_1,\ldots,H_p\}$, it is said to be $(H_1,\ldots,H_p)$-free. The class of $H$-free graphs has bounded clique-width if and only if $H$ is an induced subgraph of the 4-vertex path $P_4$. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs $H_1$ and $H_2$. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of $(H_1,H_2)$-free graphs (i) for all pairs $(H_1,H_2)$, both of which are connected, except two non-equivalent cases, and (ii) for all pairs $(H_1,H_2)$, at least one of which is not connected, except 11 non-equivalent cases. We also consider classes characterized by forbidding a finite family of graphs $\{H_1,\ldots,H_p\}$ as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colour