Graphs with forbidden subgraphs

2019

A (0, 1)-matrix has the Consecutive Ones Property (C1P) for the rows if there is a permutation of its columns such that the ones in each row appear consecutively. We say a (0, 1)-matrix is nested if it has the consecutive ones property for the rows (C1P) and every two rows are either disjoint or nested. We say a (0, 1)-matrix is 2-nested if it has the C1P and admits a partition of its rows into two sets such that the submatrix induced by each of these sets is nested. We say a split graph G with split partition (K, S) is nested (resp. 2-nested) if the matrix A(S, K) which indicates the adjacency between vertices in S and K is nested (resp. 2-nested). In this work, we characterize nested and 2-nested matrices by minimal forbidden submatrices. This characterization leads to a minimal forbidden induced subgraph characterization of these graph classes, which are superclasses of threshold graphs and subclasses of split and circle graphs. 2000 AMS Subject Classification: 05C75, 05C50 and 05C38.

Electronic Notes in Discrete Mathematics, 2018

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set S is a vertex separator for vertices a and b if the removal of S of the graph separates a and b into distinct connected components. A graph G is chordal if and only if every minimal vertex separator is a clique. We study subclasses of chordal graphs defined by restrictions imposed on the intersections of its minimal separator cliques. Our goal is to characterize them by forbidden induced subgraphs. Some of these classes have already been studied such as chordal graphs in which two minimal separators have no empty intersection if and only if they are equal. Those graphs are known as strictly chordal graphs and they were first introduced as block duplicate graphs by Golumbic and Peled [5], they were also considered in [7] and [2], showing that strictly chordal graphs are exactly the (gem, dart)-free graphs.

Combinatorica, 1985

Does there exist a function f (r, n) such that each graph G with Z(G) _-f (r, n) contains either it complete subgraph of order r or else two non-neigh boring n-chromatic s! tbgraphs? It is known that f (r, 2) exists and we establish the existence off (r, 3). We also give some interesting results about graphs %vhigh do not contain two independent edges .

Discrete Mathematics, 2002

Graphs which can be represented as nontrivial subgraphs of Cartesian product graphs are characterized. As a corollary it is shown that any bipartite, K 2,3-free graph of radius 2 has such a representation. An infinite family of graphs which have no such representation and contain no proper representable subgraph is also constructed. Only a finite number of such graphs have been previously known.

2021

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

Doklady Mathematics, 2007

European Journal of Combinatorics, 2014

The class of split graphs consists of those graphs G, for which there exist partitions (V 1 , V 2) of their vertex sets V (G) such that G[V 1 ] is an edgeless graph and G[V 2 ] is a complete graph. The classes of edgeless and complete graphs are members of a family L * ≤ , which consists of all graph classes that are induced hereditary and closed under substitution. Graph classes considered in this paper are alike to split graphs. Namely a graph class P is an object of our interest if there exist two graph classes P 1 , P 2 ∈ L * ≤ (not necessarily different) such that for each G ∈ P we can find a partition (V 1 , V 2) of V (G) satisfying G[V 1 ] ∈ P 1 and G[V 2 ] ∈ P 2. For each such class P we characterize all forbidden graphs that are strongly decomposable. The finiteness of families of forbidden graphs for P is analyzed giving a result characterizing classes with a finite number of forbidden graphs. Our investigation confirms, in the class L * ≤ , Zverovich's conjecture describing all induced hereditary graph classes defined by generalized vertex 2-partitions that have finite families of forbidden graphs.

2014

We provide a complete structural characterization of K_2,4-minor-free graphs. The 3-connected K_2,4-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains K_4 and, for each n > 5, 2n-8 nonisomorphic graphs of order n. To describe the 2-connected K_2,4-minor-free graphs we use xy-outerplanar graphs, graphs embeddable in the plane with a Hamilton xy-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, xy-outerplanar graphs are precisely the graphs that have no rooted K_2,2-minor where x and y correspond to the two vertices on one side of the bipartition of K_2,2. Each 2-connected K_2,4-minor-free graph is then (i) outerplanar, (ii) the union of three xy-outerplanar graphs and possibly the edge xy, or (iii) obtained from a 3-connected K_2,4-minor-free graph by replacing each edge x_iy_i in a set {x_1 y_1, x_2 y_2, ..., x_k y_k} satisfying a c...

Discrete Mathematics, 1999

We study the stability of some classes of claw-free graphs defined in terms of forbidden subgraphs under the closure operation defined in [10]. We characterize all connected graph.,; A such that the class of all CA-free graphs (where C denotes the claw) is stable. Using this result, we prove that every 2-connected and CHPs-free, CHZs-free or CHNt.l.4-free graph is either hamiltonian or belongs to some classes of exceptional graphs (all of them having,, connectivity 2). (~

A graph G is said to be H-regular if for each vertex v ∈ V (G), the graph induced by N G (v) is isomorphic to H. A graph H is a feasible neighbor-subgraph if there exists an H-regular graph, otherwise H is a forbidden neighbor-subgraph. In this paper, we obtain some classes of graphs H which are forbidden and then we focus on searching H-regular graphs especially those graphs of smaller order.