On graphs with no induced subdivision of K4

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 $n \ge 5$, $2n-8$ 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_...

For the class of triangle-free graphs Brooks' Theorem can be restated in terms of forbidden induced subgraphs, i.e. let G be a triangle-free and K 1,r+1-free graph. Then G is r-colourable unless G is isomorphic to an odd cycle or a complete graph with at most two vertices. In this note we present an improvement of Brooks' Theorem for triangle-free and rsunshade-free graphs. Here, an r-sunshade (with r ≥ 3) is a star K 1,r with one branch subdivided. A classical result in graph colouring theory is the theorem of Brooks [2], asserting that every graph G is (∆(G))-colourable unless G is isomorphic to an odd cycle or a complete graph. Bryant [3] simplified this proof with the following characterization of cycles and complete graphs. Thereby he highlights the exceptional role of the cycles and complete graphs in Brooks' Theorem. Here we give a new elementary proof of this characterization. Proposition 1 (Bryant [3]). Let G be a 2-connected graph. Then G is a cycle or a complete graph if and only if G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Proof. Let G be a 2-connected graph of order n. If G is a cycle or a complete graph, then obviously G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Hence, assume that G is neither a cycle nor a complete graph and that G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Note that then there exists at least one vertex v of G with 2 < d G (v) < n − 1. Since G

Graphs and Combinatorics, 2014

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| → ∞. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

Discrete Mathematics, 1993

We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class G of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members of G, the number of cliques (as well as the number of maximal independent sets) cannot exceed the number of vertices. Moreover, these graphs are almost extremal to the theorem of .

2001

Let G be a graph with n edges. It is known that there exists a cyelic Gdecomposition of K 2n+1 if and only if G has a p-Iabeling. An a-labeling of G easily yields both a cyelic G-decomposition of Kn,n and of K2nx+l for all positive integers x. It is well-known that certain classes of bipartite graphs (including certain trees) do not have a-Iabelings. :Moreover, there are bipartite graphs with n edges which do not cyclically divide Kn,n. In this article, \ve introduce the concept of an ordered p-labeling (denoted by p+) of a bipartite graph, and prove that if a graph G with n edges has a p+ -labeling, then there is a cyclic G-decomposition of K 2nx+1 for all positive integers .17. vVe also introduce the concept of a O-labeling which is a more restrictive p+ -labeling. We conjecture that all forests have a p+labeling and show that the vertex-disjoint union of any finite collection of graphs that admit a-Iabelings admits a O-labeling.

Discrete Mathematics & Theoretical Computer Science, 2009

A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size 19n/30 + 2/15 . This bound is tight, since 19n/30 − 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.

Discussiones Mathematicae Graph Theory, 2010

The edge C 4 graph of a graph G, E 4 (G) is a graph whose vertices are the edges of G and two vertices in E 4 (G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C 4. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C 4 graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E 4 (G). It is shown that for any graph G without isolated vertices, there exists a super graph H such that C(H) = G and C(E 4 (H)) = E 4 (G). Also we give forbidden subgraph characterizations for E 4 (G) being a threshold graph, block graph, geodetic graph and weakly geodetic graph.

Discrete Mathematics, 1996

The Even Pair Lemma, proved by Meyniel, states that no minimal imperfect graph contains a pair of vertices such that all chordless paths joining them have even lengths. This Lemma has proved to be very useful in the theory of perfect graphs. The Odd Pair Conjecture, with 'even' replaced by 'odd', is the natural analogue of the Even Pair Lemma. We prove a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices Ul, u2 such that all chordless paths of G joining ul to u2 contain precisely three edges. As a by-product, we obtain short proofs of two previously known theorems: the first one is a well-known theorem of Meyniel (a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords), the second one is a theorem of Olariu (a graph is perfect if it contains no odd antihole, no Ps and no extended claw as induced subgraphst.

Discrete Applied Mathematics, 1987