Journal of Graph Theory最新文献

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph $��G��\left(��n�,�r�,�d��\right)��$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

We prove that if each edge of a graph $��G�$ belongs to at least seven triangles, then $��G�$ contains a $��K_9�$-minor or contains $��K_{�1�\; �,�\; �2�\; �,�\; �2�\; �,�\; �2�\; �,�\; �2�\; �,�\; �2�}�$ as an induced subgraph. This result was conjectured by Albar and Gonçalves in 2018. Moreover, we apply this result to study the stress freeness of graphs.

A tree $��T�$ is said to be even if all pairs of vertices of degree one in $��T�$ are joined by a path of even length. We conjecture that every $��r�$-regular nonbipartite connected graph $��G�$ has a spanning even tree and verify this conjecture when $��G�$ has a 2-factor. Well-known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when $��r�$ is odd and $��G�$ has at least $��r�$ bridges. We investigate this case further and propose some related conjectures.

A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw-free, and diamond-free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of $��k�$-moplex graphs, which are defined as graphs containing at most $��k�$ moplexes. Of particular interest is the smallest nontrivial case $��k�\; �=�\; �2�$, which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity-theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected 2-moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.

Strong cocomparability graphs are the reflexive graphs whose adjacency matrix can be rearranged by a simultaneous row and column permutation to avoid the submatrix with rows $��01�\; �,�\; �10�$. Strong cocomparability graphs form a subclass of cocomparability graphs (i.e., the complements of comparability graphs) and can be recognized in polynomial time. In his seminal paper, Gallai characterized cocomparability graphs in terms of a forbidden structure called asteroids. Gallai proved that cocomparability graphs are precisely those reflexive graphs which do not contain asteroids. In this paper, we give a characterization of strong cocomparability graphs which is analogous to Gallai's characterization for cocomparability graphs. We prove that strong cocomparability graphs are precisely those reflexive graphs which do not contain weak edge-asteroids (a weaker version of asteroids). Our characterization also leads to a polynomial time recognition algorithm for strong cocomparability graphs.

We show that the maximum number of minimum dominating sets of a forest with domination number $�\; ��\gamma �$ is at most $�\; ��{\sqrt{�5�}}^{\gamma }�$ and construct for each $�\; ��\gamma �$ a tree with domination number $�\; ��\gamma �$ that has more than $�\; ��\frac{2}{5}�\; �{\sqrt{�5�}}^{\gamma }�$ minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.