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This is the first in a series of two papers dealing with $(2P_3,C_4,C_6)$-free graphs, or equivalently, $(2P_3,\text{even hole})$-free graphs. In this two-paper series, we give a full structural description of $(2P_3,C_4,C_6)$-free graphs…

A graph $G$ is called a replication graph of a graph $H$ if $G$ is obtained from $H$ by replacing vertices of $H$ by arbitrary cliques of vertices and then replacing each edge in $H$ by all the edges between corresponding cligues. For a…

A mapping $\alpha : V(G) \to V(H)$ from the vertex set of one graph $G$ to another graph $H$ is an isometric embedding if the shortest path distance between any two vertices in $G$ equals the distance between their images in $H$. Here, we…

For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if…

Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine…

A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained from $G$ after vertex deletions and edge contractions. We show that for every $k$-vertex planar graph $H$, every graph $G$ excluding $H$ as an induced minor and…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the…

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$, and a graph $G$ is perfectly weight divisible if for every…

Let $\hom(G)$ denote the size of the largest clique or independent set of a graph $G$. In 2007, Bukh and Sudakov proved that every $n$-vertex graph $G$ with $\hom(G) = O(\log n)$ contains an induced subgraph with $\Omega(n^{1/2})$ distinct…

An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as…

Extending several previous results we obtained nearly tight estimates on the maximum size of a clique-minor in various classes of expanding graphs. These results can be used to show that graphs without short cycles and other H-free graphs…

Paths $P^1,\ldots,P^k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P^i$ and $P^j$ have neither common vertices nor adjacent vertices. For a fixed integer $k$, the $k$-Induced Disjoint Paths problem is to decide if a graph…

A minimal separator of a graph $G$ is a set $S \subseteq V(G)$ such that there exist vertices $a,b \in V(G) \setminus S$ with the property that $S$ separates $a$ from $b$ in $G$, but no proper subset of $S$ does. For an integer $k\ge 0$, we…

A graph $G$ is $H$-free if it does not contain an induced subgraph isomorphic to $H$. The study of the typical structure of $H$-free graphs was initiated by Erd\H{o}s, Kleitman and Rothschild, who have shown that almost all $C_3$-free…

A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By $P_t$ we denote a path on $t$ vertices. In this paper we give polynomial-time algorithms for the following problems: the Maximum…

The class of graphs that do not contain an induced path on $k$ vertices, $P_k$-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of $P_k$-free graphs, including…

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show…

In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show…

We introduce the class of interval $H$-graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed graph $H$ with vertices $a_1,a_2,\dots,a_k$, we say that an input graph $G$ with given partition…

We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the…