Benjamin Lévêque, Frédéric Maffray, Myriam Preissmann
A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are not path graphs and are minimal with this property.
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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 that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.
We study a special kind of bounds (so called forbidden subgraph bounds, cf. Feige, Verbitsky '02) for parallel repetition of multi-prover games. First, we show that forbidden subgraph upper bounds for $r \ge 3$ provers imply the same bounds for the density Hales-Jewett theorem for alphabet of size $r$. As a consequence, this yields a new family of games with slow decrease in the parallel repetition value. Second, we introduce a new technique for proving exponential forbidden subgraph upper bounds and explore its power and limitations. In particular, we obtain exponential upper bounds for two-p…
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 minima…
A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.
A normal Helly circular-arc graph is the intersection graph of arcs on a circle of which no three or less arcs cover the whole circle. Lin, Soulignac, and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs that are not normal Helly circular-arc graphs, and used it to develop the first recognition algorithm for this graph class. As open problems, they ask for the forbidden induced subgraph characterization and a direct recognition algorithm for normal Helly circular-arc graphs, both of which are resolved by the current paper. Moreover, when the input is not a normal Helly…
It is known that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a forest whose every component is a path. In this study, we characterize all sets $\mathscr{H}$ of graphs with some $k\in \mathbb{N}$ bounding the diameter of members of $\mathscr{H}$ from above, such that $\mathscr{H}$-free graphs, i.e. graphs with no member of $\mathscr{H}$ as an induced subgraph, are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs.…
Forbidden induced subgraph characterization of circle graphs within split graphs
本文针对圆图在分裂图中的子类，首次给出了基于极小禁用导出子图的结构刻画，解决了该领域长期未决的图论问题。
We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …
We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width …
The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on $\Vec{H}$-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept ($\Vec{H}^*$-free) to get a similar conclusion about the case of undirected graph.
Word-representability and comparability: Minimal forbidden induced subgraphs and cover number bounds
研究词可表性与可比性图的最小禁止诱导子图，并确定其覆盖数界限，揭示了词可表图的覆盖数上限及特定图类的覆盖数特性。
Forbidden induced subgraphs in iterative higher order line graphs
受Beineke用9个禁用子图表征线图的结果启发，本文用纯禁用诱导子图表征二阶线图，给出高阶线图的禁用子图充分列表，并表征最大度为3和4的图的所有阶线图。
Local certification of forbidden subgraphs
本文从局部认证角度研究禁止子图检测问题，给出最优证书大小与禁止子图大小及验证局部性之间的上下界关系，并提出通用工具“分层映射”。
The edge intersection graph of a family of paths in host tree is called an $EPT$ graph. When the host tree has maximum degree $h$, we say that $G$ belongs to the class $[h,2,2]$. If, in addition, the family of paths satisfies the Helly property, then $G \in$ Helly $[h,2,2]$. The time complexity of the recognition of the classes $[h,2,2]$ inside the class $EPT$ is open for every $h> 4$. Golumbic et al. wonder if the only obstructions for an $EPT$ graph belonging to $[h,2,2]$ are the chordless cycles $C_n$ for $n> h$. In the present paper, we give a negative answer to that question, we present a…
Jones et al. (2015) introduced the notion of $u$-representable graphs, where $u$ is a word over $\{1, 2\}$ different from $22\cdots2$, as a generalization of word-representable graphs. Kitaev (2016) showed that if $u$ is of length at least 3, then every graph is $u$-representable. This indicates that there are only two nontrivial classes in the theory of $u$-representable graphs: 11-representable graphs, which correspond to word-representable graphs, and 12-representable graphs. This study deals with 12-representable graphs. Jones et al. (2015) provided a characterization of 12-representable t…
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 $c(u)\neq c(v)$ 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 prob…
Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes. By building on the work of Gioan and Paul (2012) and Chauve et al. (2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph. We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti…
A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $Δ$, then $χ(G) \leq c_F Δ/ \logΔ$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $χ(G) \leq (t + o(1)) Δ/ \logΔ$ as $Δ\to \infty$. We improve this bound to $(1+o(1)) Δ/\log Δ$, making the constant factor independent of $t$. We further extend our result …
In this paper we continue investigations of cover-incomparability graphs of finite partially ordered sets (see \cite{Bres,Bres2,Bres3,Bres4} and \cite{Max,MaxDH}). We consider in some detail the distinction between cover-preserving subsets and isometric subsets of a partially ordered set. This is critical to understanding why forbidden subposet characterizations of certain classes of cover-incomparability graphs in \cite{Bres} and \cite{Bres3} are not valid as presented. Here we provide examples, investigate the root of the difficulties, and formulate and prove valid revisions of these charact…
Computing Treedepth Obstructions
本文提出算法计算树深度不超过k的图类的禁止 minor、子图和诱导子图，针对k=4和n=16得出完整障碍集。
A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are precisely the minimal forbidden induced circular-arc subgraphs for the class of Helly circular-arc graphs. We show that it is possible to find in linear time, in any given obstacle, some minimal forbidden induced subgraph for the class of Helly circular-arc graphs contained as an induced subgraph. Moreover, relying on an existing linear-time algorithm for finding i…
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 cli…
List coloring ordered graphs with forbidden induced subgraphs
本文研究有序图中禁止单个导出子图的列表4-染色问题，给出算法与复杂性结果，形成几乎完整的二分法，仅留一个极小开放情形。
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any matrix that does not have the consecutive-ones property.
Forbidden Induced Subgraph Characterization of Word-Representable Split Graphs
本文通过引入I-圆属性，建立了半传递分裂图与邻接矩阵之间的联系，并给出了其最小禁止诱导子图的特征。
Daligault, Rao and Thomassé asked whether a hereditary class of graphs well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph $H$, as such graphs are well-quasi-ordered and are of bounded cli…
In the game of Cops and Robber, a team of cops attempts to capture a robber on a graph $G$. Initially, all cops occupy some vertices in $G$ and the robber occupies another vertex. In each round, a cop can move to one of its neighbors or stay idle, after which the robber does the same. The robber is caught by a cop if the cop lands on the same vertex which is currently occupied by the robber. The minimum number of cops needed to guarantee capture of a robber on $G$ is called the {\em cop number} of $G$, denoted by $c(G)$. We say a family $\cal F$ of graphs is {\em cop-bounded} if there is a con…
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any binary matrix that does not have the consecutive-ones property.
$k$-Leaf Powers Cannot be Characterized by a Finite Set of Forbidden Induced Subgraphs for $k \geq 5$
本文证明 $k\ge 5$ 时 $k$-叶幂图类无法通过有限禁导出子图集合在强弦图中刻画，否定了小 $k$ 情形有限刻画的推广。
The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most five vertices except 13 explicitly enumerated cases
It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering on their vertex set, and the function f(H) = sup{chi(G) | G in Forb(H)} where Forb(H) denotes the set of all ordered graphs that do not contain a copy of H. If H contains a cycle, then as in the case of unordered graphs, f(H) is infinity. However, in contrast to the unordered graphs, we describe an infinite family of ordered forests H with infinite f(H). An ordered graph is crossing if there…
Mixed Unit Interval Bigraphs : A Characterization
本文通过禁止诱导子图完整刻画了混合单位区间双图，该图类是单位区间双图的严格超类。
Forbidden Induced Subgraph Characterization of Word-Representable Co-bipartite Graphs
本文研究了词可表示共双色图的结构和算法特征，通过将共双色图与排列图联系，发现其等价于圆图，并提出基于二元矩阵圆兼容性质的半传递性特征化方法，同时给出线性时间识别算法。
An interval graph is the intersection graph of a finite set of intervals on a line and a circular-arc graph is the intersection graph of a finite set of arcs on a circle. While a forbidden induced subgraph characterization of interval graphs was found fifty years ago, finding an analogous characterization for circular-arc graphs is a long-standing open problem. In this work, we study the intersection graphs of finite sets of arcs on a circle no three of which cover the circle, known as normal Helly circular-arc graphs. Those circular-arc graphs which are minimal forbidden induced subgraphs for…
Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-graphs
本文研究仅满足有界树宽可解和三次以下图NP完全但不保持边细分NP困难性的四类问题，揭示其在禁止“H”图细分时具有不同于满足全部三条件问题的丰富复杂性边界。
Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem
本文研究$H$-子图自由图上的斯坦纳森林问题，提出四个多项式时间算法，并证明基于$c$-删除集参数化的FPT及特定情况下的多项式可解性，填补复杂性空白。
Complexity Framework For Forbidden Subgraphs I: The Framework
本研究针对HH-子图自由图提出图问题复杂性的元分类框架，给出三个判定条件，可推导问题的复杂性二分性，适用于多类图问题，还解决了若干文献开放问题。
A graph is concave-round if its vertices can be circularly enumerated so that the closed neighbourhood of each vertex is an interval in the enumeration. In this work, we give a minimal forbidden induced subgraph characterization for the class of concave-round graphs, solving a problem posed by Bang-Jensen, Huang, and Yeo [SIAM J Discrete Math, 13:179--193, 2000]. In addition, we show that it is possible to find one such forbidden induced subgraph in linear time in any given graph that is not concave-round. As part of the analysis, we obtain characterizations by minimal forbidden submatrices fo…
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…
Given a set $ξ=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $ξ$-free graph is one which does not contain any member of $% ξ$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let ${\gr{W}}_{k,ξ}$ be theexponential generating function (EGF for brief) of connected $ξ$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $ξ$, a fundamental differential recurrence satisfied by the EGFs ${\gr{W}}_{k,ξ}$ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of grap…
Constricting the Computational Complexity Gap of the $4$-Coloring Problem in $(P_t,C_3)$-free Graphs
本研究通过归约证明19≤t≤21时$(P_t,C_3)$-无图上的4着色问题为NP完全，结合计算机搜索验证归约图族的$P_{19}$-无属性，缩小了该问题复杂度未知的区间。
Path Eccentricity and Forbidden Induced Subgraphs
该研究推广了路径离心率相关结果，刻画了每个连通导出子图路径离心率小于$k$的图类，证明特定禁用子图下的图满足该性质，并提供多项式时间算法，回答了一个开放问题。
Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO
通过禁止诱导子图刻画有界灌木深度图类，证明无界灌木深度的遗传类上 MSO 强于 FO，并解决 FO 与 MSO 表达力一致的开放问题。
Edge-apexing in hereditary classes of graphs
本文证明遗传图类在边顶点化操作下保持有限禁止诱导子图性质，并针对余图类给出禁止诱导子图阶数上界为8且通过计算机搜索全部枚举。
Long induced paths in sparse graphs and graphs with forbidden patterns
该研究针对含长路径的图，通过禁止完全二部图或有序子图的限制，改进了长导出路径的下界结果，统一证明了相关结论并接近上界。
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 $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbi…
Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $(tw,ω)$-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and $k$-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor,…
Clique-free t-matchings in degree-bounded graphs
本文研究最大度不超过 t+1 的图中不含禁止完全多部子图的最大 t-匹配问题，提出基于半边小工具的快速组合算法，首次解决加权版本并加速无权版本。
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