ترغب بنشر مسار تعليمي؟ اضغط هنا

On the structure of (pan, even hole)-free graphs

71   0   0.0 ( 0 )
 نشر من قبل Steven Chaplick
 تاريخ النشر 2015
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our $O(nm)$-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our $O(n^{2.5}+nm)$-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.



قيم البحث

اقرأ أيضاً

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019]. The class ha s unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary. We prove that for every graph $G$, if $G$ excludes a fixed graph $H$ as a minor, then $G$ either has small tree-width, or $G$ contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymours excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graph have bounded tree-width [da Silva and Linhares Sales, Discrete Applied Mathematics 2010]. We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, an d more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $frac{3}{2}omega (G) -1$, and hence $chi(G)leq frac{3}{2}omega (G)$, where $omega(G)$ denotes the size of a largest clique in $G$ and $chi(G)$ denotes the chromatic number of $G$. We give an $O(nm)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O(nm)$ algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6omega(G)-1$ and clique-width at most 48.
291 - Zi-Xia Song 2021
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisi mplicial vertex. Both Hadwigers conjecture and the ErdH{o}s-Lovasz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $kge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $omega(G)<chi(G)=s+t-1$ satisfies the ErdH{o}s-Lovasz Tihany conjecture provided that $ tge s> chi(G)/3$. Furthermore, we prove that every $9$-chromatic graph $G$ with $omega(G)le 8$ has a $K_4cup K_6$ minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs.
The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS ) is a long-standing open question in even-hole-free graphs. From the hardness point of view, MIS is W[1]-hard in the class of graphs without induced 4-cycle (when parameterized by the solution size). Halfway of these, we show in this paper that MIS is FPT when parameterized by the solution size in the class of even-hole-free graphs. The main idea is to apply twice the well-known technique of augmenting graphs to extend some initial independent set.
We give an $O(n^4)$ algorithm to find a minimum clique cover of a (bull, $C_4$)-free graph, or equivalently, a minimum colouring of a (bull, $2K_2$)-free graph, where $n$ is the number of vertices of the graphs.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا