A graph in which every connected induced subgraph has a disconnected complement is called a cograph. Such graphs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We define a $2$-cograph to be a graph in which the complement of every $2$-connected induced subgraph is not $2$-connected. We show that, like cographs, $2$-cographs can be recursively defined. But, unlike cographs, $2$-cographs are closed under induced minors. We characterize the class of non-$2$-cographs for which every proper induced minor is a $2$-cograph. We further find the finitely many members of this class whose complements are also induced-minor-minimal non-$2$-cographs.
In this article the extremal structures for the mean order of connected induced subgraphs of cographs are determined. It is shown that among all connected cographs of order $n \ge 7$, the star $K_{1,n-1}$ has maximum mean connected induced subgraph order, and for $n \ge 3$, the $n$-skillet, $K_1+(K_1 \cup K_{n-2})$, has minimum mean connected induced subgraph order. It is deduced that the density for connected cographs (i.e. the ratio of the mean to the order of the graph) is asymptotically $1/2$. The mean order of all connected induced subgraphs containing a given vertex $v$ of a cograph $G$,…
A graph that can be generated from $K_1$ using joins and 0-sums is called a cograph. We define a sesquicograph to be a graph that can be generated from $K_1$ using joins, 0-sums, and 1-sums. We show that, like cographs, sesquicographs are closed under induced minors. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We obtain an analogue of this result for sesquicographs, that is, we find those non-sesquicographs for which every proper induced subgraph is a sesquicograph.
A graph is a cograph if it is $P_4$-free. A $k$-polar partition of a graph $G$ is a partition of the set of vertices of $G$ into parts $A$ and $B$ such that the subgraph induced by $A$ is a complete multipartite graph with at most $k$ parts, and the subgraph induced by $B$ is a disjoint union of at most $k$ cliques with no other edges. It is known that $k$-polar cographs can be characterized by a finite family of forbidden induced subgraphs, for any fixed $k$. A concrete family of such forbidden induced subgraphs is known for $k=1$, since $1$-polar graphs are precisely split graphs. For larger…
A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an eigenvalue in the interval $(-1,0)$. It is also shown that the multiplicity of any eigenvalue of a cograph $G$ does not exceed the sum of multiplicities of $0$ and $-1$ as eigenvalues of $G$. We introduce a partial order on the vertex set of graphs $G$ in terms of inclusions among the open and closed neighborhoods of vertices, and conjecture that the multipl…
A graph is a cograph if it does not contain a 4-vertex path as an induced subgraph. An $(s, k)$-polar partition of a graph $G$ is a partition $(A, B)$ of its vertex set such that $A$ induces a complete multipartite graph with at most $s$ parts, and $B$ induces the disjoint union of at most $k$ cliques with no other edges. A graph $G$ is said to be $(s, k)$-polar if it admits an $(s, k)$-polar partition. The concepts of $(s, \infty)$-, $(\infty, k)$-, and $(\infty, \infty)$-polar graphs can be analogously defined. Ekim, Mahadev and de Werra pioneered in the research on polar cographs, obtaining…
Let $k,l$ be nonnegative integers. A graph $G$ is $(k,l)$-polar if its vertex set admits a partition $(A,B)$ such that $A$ induces a complete multipartite graph with at most $k$ parts, and $B$ induces a disjoint union of at most $l$ cliques with no other edges. A graph is a cograph if it does not contain $P_4$ as an induced subgraph. It is known that $(k,l)$-polar cographs can be characterized through a finite family of forbidden induced subgraphs, for any fixed choice of $k$ and $l$. The problem of determining the exact members of such family for $k = 2 = l$ was posted by Ekim, Mahadev and de…
A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of $ω(G)$ and the maximum order of an induced complete multipartite subgraph of the co…
Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for the arboricity we want to partition the vertices into cycle-free subsets (i.e., forests). Arboricity is NP-hard in general, and our focus is on the arboricity of cographs. For arboricity two, we obtain the complete list of minimal cograph obstructions. These minimal obstructions do generalize to higher arboricities; however, we no longer have a complete list, …
A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph $G$ is called the Dilworth number of $G$. We prove that for any cograph $G$, the multiplicity of any eigenvalue $λ\ne0,-1$, does not exceed the Dilworth number of $G$ and show that this bound is tight. G. F. Royle [The rank of a cograph, Electron. J. Combin. 10 (2003), Note 11] p…
Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for which a greedy coloring of every induced subgraph uses only the minimally necessary number of colors $χ(G)$. In recent work on reciprocal best match graphs so-called hierarchically coloring play an important role. Here we show that greedy colorings are a special case of hierarchical coloring, which also require no more than $χ(G)$ colors.
Given a simple graph $G$, a set $C \subseteq V(G)$ is a neighborhood cover set if every edge and vertex of $G$ belongs to some $G[v]$ with $v \in C$, where $G[v]$ denotes the subgraph of $G$ induced by the closed neighborhood of the vertex $v$. Two elements of $E(G) \cup V(G)$ are neighborhood-independent if there is no vertex $v\in V(G)$ such that both elements are in $G[v]$. A set $S\subseteq V(G)\cup E(G)$ is neighborhood-independent if every pair of elements of $S$ is neighborhood-independent. Let $ρ_{\mathrm n}(G)$ be the size of a minimum neighborhood cover set and $α_{\mathrm n}(G)$ of …
For a pair of natural numbers $k, l$, a $(k,l)$-colouring of a graph $G$ is a partition of the vertex set of $G$ into (possibly empty) sets $S_1, S_2, \dots, S_k$, $C_1, C_2, \dots, C_l$ such that each set $S_i$ is an independent set and each set $C_j$ induces a clique in $G$. The $(k,l)$-colouring problem, which is NP-complete in general, has been studied for special graph classes such as chordal graphs, cographs and line graphs. Let $\hatκ(G) = (κ_0(G),κ_1(G),\dots,κ_{θ(G)-1}(G))$ and $\hatλ(G) = (λ_0(G),λ_1(G),\dots,λ_{χ(G)-1}(G))$ where $κ_l(G)$ (respectively, $λ_k(G)$) is the minimum $k$ …
Zero forcing is a process that models the spread of information throughout a graph as white vertices are forced to turn blue using a color change rule. The idea of throttling, introduced in 2013 by Butler and Young, is to optimize the trade-off between the number of initial blue vertices and the time taken to force all vertices to become blue. The original throttling number of a graph minimizes the sum of these two quantities and the product throttling number minimizes their product. In addition, weighted throttling changes the weights given to these two quantities when minimizing their sum. S…
An interval graph is considered improper if and only if it has a representation such that an interval contains another interval. Previously these have been investigated in terms of balance and minimal forbidden interval subgraphs for the class of 1-improper interval graphs. This paper investigates the minimal forbidden interval sub-graphs further, generalizing results to all p-improper interval graphs. It is apparent that there are many different types of possible minimal forbidden subgraphs that fall into four broad categories.
Let $γ_g(G)$ and $γ_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $γ_g$-perfect (resp. $γ_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $γ_g(F)=γ(F)$ (resp. $γ_{tg}(F)=γ_t(F)$). A recursive characterization of $γ_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $γ_g$-perfect graphs. It is proved that every minimally $γ_g$-imperfect graph has domination number $2$. All minimally $γ_g$-imperfect triangle-free graphs are determined. It is also proved that $γ_{tg…
In connection with his solution of the Sensitivity Conjecture, Hao Huang (arXiv: 1907.00847, 2019) asked the following question: Given a graph $G$ with high symmetry, what can we say about the smallest maximum degree of induced subgraphs of $G$ with $α(G)+1$ vertices, where $α(G)$ denotes the size of the largest independent set in $G$? We study this question for $H(n,k)$, the $n$-dimensional Hamming graph over an alphabet of size $k$. Generalizing a construction by Chung et al. (JCT-A, 1988), we prove that $H(n,k)$ has an induced subgraph with more than $α(H(n,k))$ vertices and maximum degree …
Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attentio…
A cycle-transversal of a graph G is a subset T of V(G) such that T intersects every cycle of G. A clique cycle-transversal, or cct for short, is a cycle-transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distance-hereditary graphs admitting a cct in terms of forbidden induced subgraphs. This extends similar results for chordal graphs and cographs.
Cographs are exactly the hereditarily well-colored graphs, i.e., the graphs for which a greedy vertex coloring of every induced subgraph uses only the minimally necessary number of colors $χ(G)$. We show that greedy colorings are a special case of the more general hierarchical vertex colorings, which recently were introduced in phylogenetic combinatorics. Replacing cotrees by modular decomposition trees generalizes the concept of hierarchical colorings to arbitrary graphs. We show that every graph has a modularly-minimal coloring $σ$ satisfying $|σ(M)|=χ(M)$ for every strong module $M$ of $G$.…
The Kneser graph ${\rm KG}_{n,k}$ is a graph whose vertex set is the family of all $k$-subsets of $[n]$ and two vertices are adjacent if their corresponding subsets are disjoint. The classical Erdős-Ko-Rado theorem determines the cardinality and structure of a maximum induced $K_2$-free subgraph in ${\rm KG}_{n,k}$. As a generalization of the Erdős-Ko-Rado theorem, Erdős proposed a conjecture about the maximum order of an induced $K_{s+1}$-free subgraph of ${\rm KG}_{n,k}$. As the best known result concerning this conjecture, Frankl [Journal of Combinatorial Theory, Series A, 2013], when $n\ge…
We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turan, Erdos-Simonovits and Chung-Graham-Wilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractal-like) structure. We also give some necessary conditions for forcibility, which impl…
Let $Γ_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $Γ_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. The…
Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Győri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove that $c=\frac{3}{8}$. More generally, we show that for any $\varepsilon>0$, and any integer $k \ge 2$, there is a $C_{2k}$-free graph $G_1$ which does not contain a bipartite subgraph of girth greater than $2k$ with more than $\left(1-\frac{1}{2^{2k-2}}\right)\frac{2}{2k-1}(1+\varepsilon)$ fraction of the edges of $G_1$. There also exists a $C_{2k}$-free graph $G_2$ whi…
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $Δ(G)>|V(G)|/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. The 1-factorization conjecture is a special case of this overfull conjecture, which states that for even $n$, every regular $n$-vertex graph with degree at least about $n/2$ has a 1-factorization and was confirmed for large graphs in 2014. Supporting the overfull conjecture as well as generalizing the 1-factorizat…
In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs $G$ of Cartesian products $G_1\times\cdots\times G_m$ of arbitrary connected graphs. Namely, we show that $\frac{|E(G)|}{|V(G)|}\le \lceil 2\max\{ \text{dens}(G_1),\ldots,\text{dens}(G_m)\} \rceil\log|V(G)|$, where $\text{dens}(H)$ is the maximum ratio $\frac{|E(H')|}{|V(H')|}$ over all subgraphs $H'$ of $H$. We introduce the notions of VC-dimension $\text{VC-dim}(G)$ and VC-density $\text{VC-dens}(G)$ of a subgraph $G$ of a Cartesian product $G_1\times\cdots\times G_m$, generalizing the cl…
It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a…
A family of graphs $\mathcal{F}$ is hereditary if $\mathcal{F}$ is closed under isomorphism and taking induced subgraphs. The speed of $\mathcal{F}$ is the sequence $\{|\mathcal{F}^n|\}_{n \in \mathbb{N}}$, where $\mathcal{F}^n$ denotes the set of graphs in $\mathcal{F}$ with the vertex set $[n]$. Alon, Balogh, Bollobás and Morris [The structure of almost all graphs in a hereditary property, JCTB 2011] gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family $\mathcal{F}$ there exist $\varepsilon>0$ and an integer $l \geq 1$ such …
We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on…
A graph is $H$-free if it contains no induced subgraph isomorphic to $H$. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on $(sP_1+P_3)$-free graphs for every integer $s\geq 1$. We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these r…
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph c…
A Complete Proof of the Strong Conjecture about $F$-Irregular Graphs
Research Laboratory "Mathematics of Hybrid Intelligence Systems", Francisk Skorina Gomel State University（弗朗西斯·斯科里纳戈梅尔国立大学混合智能系统数学研究实验室）；Faculty of Applied Mathematics and Computer Science, Belarusian State University（白俄罗斯国立大学应用数学与计算机科学系）
For a fixed graph $H$, the $H$-IS-Deletion problem asks, given a graph $G$, for the minimum size of a set $S \subseteq V(G)$ such that $G\setminus S$ does not contain $H$ as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph $H$, the smallest function $f_H(t)$ such that $H$-IS-Deletion can be solved in time $f_H(t) \cdot n^{O(1)}$ assuming the Exponential Time Hypothesis (ETH), where $t$ and $n$ denote the treewidth and the number of vertices of the input graph, respec…
